doubleOTHERcomputational(3)

NAME
doubleOTHERcomputational - double

SYNOPSIS

  Functions
subroutine ctplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
CTPLQT
subroutine ctplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
CTPLQT2
subroutine ctpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda, b, ldb, work, info)
CTPMLQT
subroutine dbbcsd (jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b22d, b22e, work, lwork, info)
DBBCSD
subroutine dgetsqrhrt (m, n, mb1, nb1, nb2, a, lda, t, ldt, work, lwork, info)
DGETSQRHRT
subroutine dgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
DGGHD3
subroutine dgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
DGGHRD
subroutine dggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
DGGQRF
subroutine dggrqf (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
DGGRQF
subroutine dggsvp3 (jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, tau, work, lwork, info)
DGGSVP3
subroutine dgsvj0 (jobv, m, n, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info)
DGSVJ0 pre-processor for the routine dgesvj.
subroutine dgsvj1 (jobv, m, n, n1, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info)
DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots.
subroutine dhsein (side, eigsrc, initv, select, n, h, ldh, wr, wi, vl, ldvl, vr, ldvr, mm, m, work, ifaill, ifailr, info)
DHSEIN
subroutine dhseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
DHSEQR
subroutine dla_lin_berr (n, nz, nrhs, res, ayb, berr)
DLA_LIN_BERR computes a component-wise relative backward error.
subroutine dla_wwaddw (n, x, y, w)
DLA_WWADDW adds a vector into a doubled-single vector.
subroutine dlals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info)
DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine dlalsa (icompq, smlsiz, n, nrhs, b, ldb, bx, ldbx, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info)
DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine dlalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info)
DLALSD uses the singular value decomposition of A to solve the least squares problem.
double precision function dlansf (norm, transr, uplo, n, a, work)
DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.
subroutine dlarscl2 (m, n, d, x, ldx)
DLARSCL2 performs reciprocal diagonal scaling on a matrix.
subroutine dlarz (side, m, n, l, v, incv, tau, c, ldc, work)
DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
subroutine dlarzb (side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, c, ldc, work, ldwork)
DLARZB applies a block reflector or its transpose to a general matrix.
subroutine dlarzt (direct, storev, n, k, v, ldv, tau, t, ldt)
DLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine dlascl2 (m, n, d, x, ldx)
DLASCL2 performs diagonal scaling on a matrix.
subroutine dlatrz (m, n, l, a, lda, tau, work)
DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine dopgtr (uplo, n, ap, tau, q, ldq, work, info)
DOPGTR
subroutine dopmtr (side, uplo, trans, m, n, ap, tau, c, ldc, work, info)
DOPMTR
subroutine dorbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
DORBDB
subroutine dorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
DORBDB1
subroutine dorbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
DORBDB2
subroutine dorbdb3 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
DORBDB3
subroutine dorbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
DORBDB4
subroutine dorbdb5 (m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
DORBDB5
subroutine dorbdb6 (m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
DORBDB6
recursive subroutine dorcsd (jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, iwork, info)
DORCSD
subroutine dorcsd2by1 (jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, iwork, info)
DORCSD2BY1
subroutine dorg2l (m, n, k, a, lda, tau, work, info)
DORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine dorg2r (m, n, k, a, lda, tau, work, info)
DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine dorghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
DORGHR
subroutine dorgl2 (m, n, k, a, lda, tau, work, info)
DORGL2
subroutine dorglq (m, n, k, a, lda, tau, work, lwork, info)
DORGLQ
subroutine dorgql (m, n, k, a, lda, tau, work, lwork, info)
DORGQL
subroutine dorgqr (m, n, k, a, lda, tau, work, lwork, info)
DORGQR
subroutine dorgr2 (m, n, k, a, lda, tau, work, info)
DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
subroutine dorgrq (m, n, k, a, lda, tau, work, lwork, info)
DORGRQ
subroutine dorgtr (uplo, n, a, lda, tau, work, lwork, info)
DORGTR
subroutine dorgtsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
DORGTSQR
subroutine dorgtsqr_row (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
DORGTSQR_ROW
subroutine dorhr_col (m, n, nb, a, lda, t, ldt, d, info)
DORHR_COL
subroutine dorm2l (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine dorm2r (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine dormbr (vect, side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMBR
subroutine dormhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
DORMHR
subroutine dorml2 (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
subroutine dormlq (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMLQ
subroutine dormql (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMQL
subroutine dormqr (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMQR
subroutine dormr2 (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
subroutine dormr3 (side, trans, m, n, k, l, a, lda, tau, c, ldc, work, info)
DORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
subroutine dormrq (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMRQ
subroutine dormrz (side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info)
DORMRZ
subroutine dormtr (side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info)
DORMTR
subroutine dpbcon (uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info)
DPBCON
subroutine dpbequ (uplo, n, kd, ab, ldab, s, scond, amax, info)
DPBEQU
subroutine dpbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DPBRFS
subroutine dpbstf (uplo, n, kd, ab, ldab, info)
DPBSTF
subroutine dpbtf2 (uplo, n, kd, ab, ldab, info)
DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
subroutine dpbtrf (uplo, n, kd, ab, ldab, info)
DPBTRF
subroutine dpbtrs (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
DPBTRS
subroutine dpftrf (transr, uplo, n, a, info)
DPFTRF
subroutine dpftri (transr, uplo, n, a, info)
DPFTRI
subroutine dpftrs (transr, uplo, n, nrhs, a, b, ldb, info)
DPFTRS
subroutine dppcon (uplo, n, ap, anorm, rcond, work, iwork, info)
DPPCON
subroutine dppequ (uplo, n, ap, s, scond, amax, info)
DPPEQU
subroutine dpprfs (uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DPPRFS
subroutine dpptrf (uplo, n, ap, info)
DPPTRF
subroutine dpptri (uplo, n, ap, info)
DPPTRI
subroutine dpptrs (uplo, n, nrhs, ap, b, ldb, info)
DPPTRS
subroutine dpstf2 (uplo, n, a, lda, piv, rank, tol, work, info)
DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine dpstrf (uplo, n, a, lda, piv, rank, tol, work, info)
DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine dsbgst (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)
DSBGST
subroutine dsbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
DSBTRD
subroutine dsfrk (transr, uplo, trans, n, k, alpha, a, lda, beta, c)
DSFRK performs a symmetric rank-k operation for matrix in RFP format.
subroutine dspcon (uplo, n, ap, ipiv, anorm, rcond, work, iwork, info)
DSPCON
subroutine dspgst (itype, uplo, n, ap, bp, info)
DSPGST
subroutine dsprfs (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DSPRFS
subroutine dsptrd (uplo, n, ap, d, e, tau, info)
DSPTRD
subroutine dsptrf (uplo, n, ap, ipiv, info)
DSPTRF
subroutine dsptri (uplo, n, ap, ipiv, work, info)
DSPTRI
subroutine dsptrs (uplo, n, nrhs, ap, ipiv, b, ldb, info)
DSPTRS
subroutine dstegr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
DSTEGR
subroutine dstein (n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
DSTEIN
subroutine dstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
DSTEMR
subroutine dtbcon (norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info)
DTBCON
subroutine dtbrfs (uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DTBRFS
subroutine dtbtrs (uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info)
DTBTRS
subroutine dtfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine dtftri (transr, uplo, diag, n, a, info)
DTFTRI
subroutine dtfttp (transr, uplo, n, arf, ap, info)
DTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
subroutine dtfttr (transr, uplo, n, arf, a, lda, info)
DTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
subroutine dtgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
DTGSEN
subroutine dtgsja (jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info)
DTGSJA
subroutine dtgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
DTGSNA
subroutine dtpcon (norm, uplo, diag, n, ap, rcond, work, iwork, info)
DTPCON
subroutine dtplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
DTPLQT
subroutine dtplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
DTPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine dtpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda, b, ldb, work, info)
DTPMLQT
subroutine dtpmqrt (side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info)
DTPMQRT
subroutine dtpqrt (m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)
DTPQRT
subroutine dtpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
DTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine dtprfs (uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DTPRFS
subroutine dtptri (uplo, diag, n, ap, info)
DTPTRI
subroutine dtptrs (uplo, trans, diag, n, nrhs, ap, b, ldb, info)
DTPTRS
subroutine dtpttf (transr, uplo, n, ap, arf, info)
DTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
subroutine dtpttr (uplo, n, ap, a, lda, info)
DTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
subroutine dtrcon (norm, uplo, diag, n, a, lda, rcond, work, iwork, info)
DTRCON
subroutine dtrevc (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info)
DTREVC
subroutine dtrevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)
DTREVC3
subroutine dtrexc (compq, n, t, ldt, q, ldq, ifst, ilst, work, info)
DTREXC
subroutine dtrrfs (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info)
DTRRFS
subroutine dtrsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)
DTRSEN
subroutine dtrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)
DTRSNA
subroutine dtrti2 (uplo, diag, n, a, lda, info)
DTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
subroutine dtrtri (uplo, diag, n, a, lda, info)
DTRTRI
subroutine dtrtrs (uplo, trans, diag, n, nrhs, a, lda, b, ldb, info)
DTRTRS
subroutine dtrttf (transr, uplo, n, a, lda, arf, info)
DTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
subroutine dtrttp (uplo, n, a, lda, ap, info)
DTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
subroutine dtzrzf (m, n, a, lda, tau, work, lwork, info)
DTZRZF
subroutine stplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
STPLQT
subroutine stplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
STPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine stpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda, b, ldb, work, info)
STPMLQT
subroutine ztplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
ZTPLQT
subroutine ztplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
ZTPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine ztpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda, b, ldb, work, info)
ZTPMLQT

Detailed Description

This is the group of double other Computational routines

Function Documentation

  subroutine ctplqt (integer m, integer n, integer l, integer mb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) work, integer info)

CTPLQT

Purpose:

     CTPLQT computes a blocked LQ factorization of a complex
     'triangular-pentagonal' matrix C, which is composed of a
     triangular block A and pentagonal block B, using the compact
     WY representation for Q.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix B, and the order of the
              triangular matrix A.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size to be used in the blocked QR.  M >= MB >= 1.

    A

              A is COMPLEX array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is COMPLEX array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is COMPLEX array, dimension (LDT,N)
              The lower triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.  See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

              WORK is COMPLEX array, dimension (MB*M)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ] [ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
      upper trapezoidal matrix B2:
              [ B ] = [ B1 ] [ B2 ]
                       [ B1 ]  <- M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                [ C ] = [ A ] [ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                [ W ] = [ I ] [ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                [ V ] = [ V1 ] [ V2 ]
                       [ V1 ] <- M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The number of blocks is B = ceiling(M/MB), where each
      block is of order MB except for the last block, which is of order
      IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-N matrix T as
                   T = [T1 T2 ... TB].

Definition at line 172 of file ctplqt.f.

  subroutine ctplqt2 (integer m, integer n, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldt, * ) t, integer ldt, integer info)

CTPLQT2

Purpose:

     CTPLQT2 computes a LQ a factorization of a complex 'triangular-pentagonal'
     matrix C, which is composed of a triangular block A and pentagonal block B,
     using the compact WY representation for Q.

Parameters
    M

              M is INTEGER
              The total number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B, and the order of
              the triangular matrix A.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    A

              A is COMPLEX array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is COMPLEX array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is COMPLEX array, dimension (LDT,M)
              The N-by-N upper triangular factor T of the block reflector.
              See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= max(1,M)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ][ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
      upper trapezoidal matrix B2:
                   B = [ B1 ][ B2 ]
                       [ B1 ]  <-     M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                   C = [ A ][ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                   W = [ I ][ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                   W = [ V1 ][ V2 ]
                       [ V1 ] <-     M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The (M+N)-by-(M+N) block reflector H is then given by
                   H = I - W**T * T * W
      where W^H is the conjugate transpose of W and T is the upper triangular
      factor of the block reflector.

Definition at line 161 of file ctplqt2.f.

  subroutine ctpmlqt (character side, character trans, integer m, integer n, integer k, integer l, integer mb, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer info)

CTPMLQT

Purpose:

     CTPMLQT applies a complex unitary matrix Q obtained from a
     'triangular-pentagonal' complex block reflector H to a general
     complex matrix C, which consists of two blocks A and B.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**H from the Left;
              = 'R': apply Q or Q**H from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'C':  Conjugate transpose, apply Q**H.

    M

              M is INTEGER
              The number of rows of the matrix B. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.

    L

              L is INTEGER
              The order of the trapezoidal part of V.
              K >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size used for the storage of T.  K >= MB >= 1.
              This must be the same value of MB used to generate T
              in CTPLQT.

    V

              V is COMPLEX array, dimension (LDV,K)
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              CTPLQT in B.  See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= K.

    T

              T is COMPLEX array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by CTPLQT, stored as a MB-by-K matrix.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    A

              A is COMPLEX array, dimension
              (LDA,N) if SIDE = 'L' or
              (LDA,K) if SIDE = 'R'
              On entry, the K-by-N or M-by-K matrix A.
              On exit, A is overwritten by the corresponding block of
              Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,K);
              If SIDE = 'R', LDA >= max(1,M).

    B

              B is COMPLEX array, dimension (LDB,N)
              On entry, the M-by-N matrix B.
              On exit, B is overwritten by the corresponding block of
              Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.
              LDB >= max(1,M).

    WORK

              WORK is COMPLEX array. The dimension of WORK is
               N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The columns of the pentagonal matrix V contain the elementary reflectors
      H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
      trapezoidal block V2:
            V = [V1] [V2].
      The size of the trapezoidal block V2 is determined by the parameter L,
      where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L
      rows of a K-by-K upper triangular matrix.  If L=K, V2 is lower triangular;
      if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
      If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                          [B]
      If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.
      The complex unitary matrix Q is formed from V and T.
      If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
      If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C.
      If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
      If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

Definition at line 197 of file ctpmlqt.f.

  subroutine dbbcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, integer m, integer p, integer q, double precision, dimension( * ) theta, double precision, dimension( * ) phi, double precision, dimension( ldu1, * ) u1, integer ldu1, double precision, dimension( ldu2, * ) u2, integer ldu2, double precision, dimension( ldv1t, * ) v1t, integer ldv1t, double precision, dimension( ldv2t, * ) v2t, integer ldv2t, double precision, dimension( * ) b11d, double precision, dimension( * ) b11e, double precision, dimension( * ) b12d, double precision, dimension( * ) b12e, double precision, dimension( * ) b21d, double precision, dimension( * ) b21e, double precision, dimension( * ) b22d, double precision, dimension( * ) b22e, double precision, dimension( * ) work, integer lwork, integer info)

DBBCSD

Purpose:

     DBBCSD computes the CS decomposition of an orthogonal matrix in
     bidiagonal-block form,
         [ B11 | B12 0  0 ]
         [  0  |  0 -I  0 ]
     X = [----------------]
         [ B21 | B22 0  0 ]
         [  0  |  0  0  I ]
                                   [  C | -S  0  0 ]
                       [ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**T
                     = [---------] [---------------] [---------]   .
                       [    | U2 ] [  S |  C  0  0 ] [    | V2 ]
                                   [  0 |  0  0  I ]
     X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
     than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
     transposed and/or permuted. This can be done in constant time using
     the TRANS and SIGNS options. See DORCSD for details.)
     The bidiagonal matrices B11, B12, B21, and B22 are represented
     implicitly by angles THETA(1:Q) and PHI(1:Q-1).
     The orthogonal matrices U1, U2, V1T, and V2T are input/output.
     The input matrices are pre- or post-multiplied by the appropriate
     singular vector matrices.

Parameters
    JOBU1

              JOBU1 is CHARACTER
              = 'Y':      U1 is updated;
              otherwise:  U1 is not updated.

    JOBU2

              JOBU2 is CHARACTER
              = 'Y':      U2 is updated;
              otherwise:  U2 is not updated.

    JOBV1T

              JOBV1T is CHARACTER
              = 'Y':      V1T is updated;
              otherwise:  V1T is not updated.

    JOBV2T

              JOBV2T is CHARACTER
              = 'Y':      V2T is updated;
              otherwise:  V2T is not updated.

    TRANS

              TRANS is CHARACTER
              = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                          order;
              otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                          major order.

    M

              M is INTEGER
              The number of rows and columns in X, the orthogonal matrix in
              bidiagonal-block form.

    P

              P is INTEGER
              The number of rows in the top-left block of X. 0 <= P <= M.

    Q

              Q is INTEGER
              The number of columns in the top-left block of X.
              0 <= Q <= MIN(P,M-P,M-Q).

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
              On entry, the angles THETA(1),...,THETA(Q) that, along with
              PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
              form. On exit, the angles whose cosines and sines define the
              diagonal blocks in the CS decomposition.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
              The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
              THETA(Q), define the matrix in bidiagonal-block form.

    U1

              U1 is DOUBLE PRECISION array, dimension (LDU1,P)
              On entry, a P-by-P matrix. On exit, U1 is postmultiplied
              by the left singular vector matrix common to [ B11 ; 0 ] and
              [ B12 0 0 ; 0 -I 0 0 ].

    LDU1

              LDU1 is INTEGER
              The leading dimension of the array U1, LDU1 >= MAX(1,P).

    U2

              U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
              On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
              postmultiplied by the left singular vector matrix common to
              [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].

    LDU2

              LDU2 is INTEGER
              The leading dimension of the array U2, LDU2 >= MAX(1,M-P).

    V1T

              V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
              On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
              by the transpose of the right singular vector
              matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].

    LDV1T

              LDV1T is INTEGER
              The leading dimension of the array V1T, LDV1T >= MAX(1,Q).

    V2T

              V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
              On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
              premultiplied by the transpose of the right
              singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
              [ B22 0 0 ; 0 0 I ].

    LDV2T

              LDV2T is INTEGER
              The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).

    B11D

              B11D is DOUBLE PRECISION array, dimension (Q)
              When DBBCSD converges, B11D contains the cosines of THETA(1),
              ..., THETA(Q). If DBBCSD fails to converge, then B11D
              contains the diagonal of the partially reduced top-left
              block.

    B11E

              B11E is DOUBLE PRECISION array, dimension (Q-1)
              When DBBCSD converges, B11E contains zeros. If DBBCSD fails
              to converge, then B11E contains the superdiagonal of the
              partially reduced top-left block.

    B12D

              B12D is DOUBLE PRECISION array, dimension (Q)
              When DBBCSD converges, B12D contains the negative sines of
              THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
              B12D contains the diagonal of the partially reduced top-right
              block.

    B12E

              B12E is DOUBLE PRECISION array, dimension (Q-1)
              When DBBCSD converges, B12E contains zeros. If DBBCSD fails
              to converge, then B12E contains the subdiagonal of the
              partially reduced top-right block.

    B21D

              B21D is DOUBLE PRECISION  array, dimension (Q)
              When DBBCSD converges, B21D contains the negative sines of
              THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
              B21D contains the diagonal of the partially reduced bottom-left
              block.

    B21E

              B21E is DOUBLE PRECISION  array, dimension (Q-1)
              When DBBCSD converges, B21E contains zeros. If DBBCSD fails
              to converge, then B21E contains the subdiagonal of the
              partially reduced bottom-left block.

    B22D

              B22D is DOUBLE PRECISION  array, dimension (Q)
              When DBBCSD converges, B22D contains the negative sines of
              THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
              B22D contains the diagonal of the partially reduced bottom-right
              block.

    B22E

              B22E is DOUBLE PRECISION  array, dimension (Q-1)
              When DBBCSD converges, B22E contains zeros. If DBBCSD fails
              to converge, then B22E contains the subdiagonal of the
              partially reduced bottom-right block.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= MAX(1,8*Q).
              If LWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the WORK array,
              returns this value as the first entry of the work array, and
              no error message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if DBBCSD did not converge, INFO specifies the number
                    of nonzero entries in PHI, and B11D, B11E, etc.,
                    contain the partially reduced matrix.

Internal Parameters:

      TOLMUL  DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
              TOLMUL controls the convergence criterion of the QR loop.
              Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
              are within TOLMUL*EPS of either bound.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 328 of file dbbcsd.f.

  subroutine dgetsqrhrt (integer m, integer n, integer mb1, integer nb1, integer nb2, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer lwork, integer info)

DGETSQRHRT

Purpose:

     DGETSQRHRT computes a NB2-sized column blocked QR-factorization
     of a real M-by-N matrix A with M >= N,
        A = Q * R.
     The routine uses internally a NB1-sized column blocked and MB1-sized
     row blocked TSQR-factorization and perfors the reconstruction
     of the Householder vectors from the TSQR output. The routine also
     converts the R_tsqr factor from the TSQR-factorization output into
     the R factor that corresponds to the Householder QR-factorization,
        A = Q_tsqr * R_tsqr = Q * R.
     The output Q and R factors are stored in the same format as in DGEQRT
     (Q is in blocked compact WY-representation). See the documentation
     of DGEQRT for more details on the format.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. M >= N >= 0.

    MB1

              MB1 is INTEGER
              The row block size to be used in the blocked TSQR.
              MB1 > N.

    NB1

              NB1 is INTEGER
              The column block size to be used in the blocked TSQR.
              N >= NB1 >= 1.

    NB2

              NB2 is INTEGER
              The block size to be used in the blocked QR that is
              output. NB2 >= 1.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry: an M-by-N matrix A.
              On exit:
               a) the elements on and above the diagonal
                  of the array contain the N-by-N upper-triangular
                  matrix R corresponding to the Householder QR;
               b) the elements below the diagonal represent Q by
                  the columns of blocked V (compact WY-representation).

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    T

              T is DOUBLE PRECISION array, dimension (LDT,N))
              The upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= NB2.

    WORK

              (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              The dimension of the array WORK.
              LWORK >= MAX( LWT + LW1, MAX( LWT+N*N+LW2, LWT+N*N+N ) ),
              where
                 NUM_ALL_ROW_BLOCKS = CEIL((M-N)/(MB1-N)),
                 NB1LOCAL = MIN(NB1,N).
                 LWT = NUM_ALL_ROW_BLOCKS * N * NB1LOCAL,
                 LW1 = NB1LOCAL * N,
                 LW2 = NB1LOCAL * MAX( NB1LOCAL, ( N - NB1LOCAL ) ),
              If LWORK = -1, then a workspace query is assumed.
              The routine only calculates the optimal size of the WORK
              array, returns this value as the first entry of the WORK
              array, and no error message related to LWORK is issued
              by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

     November 2020, Igor Kozachenko,
                    Computer Science Division,
                    University of California, Berkeley

Definition at line 177 of file dgetsqrhrt.f.

  subroutine dgghd3 (character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)

DGGHD3

Purpose:

     DGGHD3 reduces a pair of real matrices (A,B) to generalized upper
     Hessenberg form using orthogonal transformations, where A is a
     general matrix and B is upper triangular.  The form of the
     generalized eigenvalue problem is
        A*x = lambda*B*x,
     and B is typically made upper triangular by computing its QR
     factorization and moving the orthogonal matrix Q to the left side
     of the equation.
     This subroutine simultaneously reduces A to a Hessenberg matrix H:
        Q**T*A*Z = H
     and transforms B to another upper triangular matrix T:
        Q**T*B*Z = T
     in order to reduce the problem to its standard form
        H*y = lambda*T*y
     where y = Z**T*x.
     The orthogonal matrices Q and Z are determined as products of Givens
     rotations.  They may either be formed explicitly, or they may be
     postmultiplied into input matrices Q1 and Z1, so that
          Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
          Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
     If Q1 is the orthogonal matrix from the QR factorization of B in the
     original equation A*x = lambda*B*x, then DGGHD3 reduces the original
     problem to generalized Hessenberg form.
     This is a blocked variant of DGGHRD, using matrix-matrix
     multiplications for parts of the computation to enhance performance.

Parameters
    COMPQ

              COMPQ is CHARACTER*1
              = 'N': do not compute Q;
              = 'I': Q is initialized to the unit matrix, and the
                     orthogonal matrix Q is returned;
              = 'V': Q must contain an orthogonal matrix Q1 on entry,
                     and the product Q1*Q is returned.

    COMPZ

              COMPZ is CHARACTER*1
              = 'N': do not compute Z;
              = 'I': Z is initialized to the unit matrix, and the
                     orthogonal matrix Z is returned;
              = 'V': Z must contain an orthogonal matrix Z1 on entry,
                     and the product Z1*Z is returned.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI mark the rows and columns of A which are to be
              reduced.  It is assumed that A is already upper triangular
              in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
              normally set by a previous call to DGGBAL; otherwise they
              should be set to 1 and N respectively.
              1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

    A

              A is DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the N-by-N general matrix to be reduced.
              On exit, the upper triangle and the first subdiagonal of A
              are overwritten with the upper Hessenberg matrix H, and the
              rest is set to zero.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the N-by-N upper triangular matrix B.
              On exit, the upper triangular matrix T = Q**T B Z.  The
              elements below the diagonal are set to zero.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the orthogonal matrix Q1,
              typically from the QR factorization of B.
              On exit, if COMPQ='I', the orthogonal matrix Q, and if
              COMPQ = 'V', the product Q1*Q.
              Not referenced if COMPQ='N'.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the orthogonal matrix Z1.
              On exit, if COMPZ='I', the orthogonal matrix Z, and if
              COMPZ = 'V', the product Z1*Z.
              Not referenced if COMPZ='N'.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.
              LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= 1.
              For optimum performance LWORK >= 6*N*NB, where NB is the
              optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      This routine reduces A to Hessenberg form and maintains B in triangular form
      using a blocked variant of Moler and Stewart's original algorithm,
      as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
      (BIT 2008).

Definition at line 228 of file dgghd3.f.

  subroutine dgghrd (character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, integer info)

DGGHRD

Purpose:

     DGGHRD reduces a pair of real matrices (A,B) to generalized upper
     Hessenberg form using orthogonal transformations, where A is a
     general matrix and B is upper triangular.  The form of the
     generalized eigenvalue problem is
        A*x = lambda*B*x,
     and B is typically made upper triangular by computing its QR
     factorization and moving the orthogonal matrix Q to the left side
     of the equation.
     This subroutine simultaneously reduces A to a Hessenberg matrix H:
        Q**T*A*Z = H
     and transforms B to another upper triangular matrix T:
        Q**T*B*Z = T
     in order to reduce the problem to its standard form
        H*y = lambda*T*y
     where y = Z**T*x.
     The orthogonal matrices Q and Z are determined as products of Givens
     rotations.  They may either be formed explicitly, or they may be
     postmultiplied into input matrices Q1 and Z1, so that
          Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
          Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
     If Q1 is the orthogonal matrix from the QR factorization of B in the
     original equation A*x = lambda*B*x, then DGGHRD reduces the original
     problem to generalized Hessenberg form.

Parameters
    COMPQ

              COMPQ is CHARACTER*1
              = 'N': do not compute Q;
              = 'I': Q is initialized to the unit matrix, and the
                     orthogonal matrix Q is returned;
              = 'V': Q must contain an orthogonal matrix Q1 on entry,
                     and the product Q1*Q is returned.

    COMPZ

              COMPZ is CHARACTER*1
              = 'N': do not compute Z;
              = 'I': Z is initialized to the unit matrix, and the
                     orthogonal matrix Z is returned;
              = 'V': Z must contain an orthogonal matrix Z1 on entry,
                     and the product Z1*Z is returned.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI mark the rows and columns of A which are to be
              reduced.  It is assumed that A is already upper triangular
              in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
              normally set by a previous call to DGGBAL; otherwise they
              should be set to 1 and N respectively.
              1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

    A

              A is DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the N-by-N general matrix to be reduced.
              On exit, the upper triangle and the first subdiagonal of A
              are overwritten with the upper Hessenberg matrix H, and the
              rest is set to zero.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the N-by-N upper triangular matrix B.
              On exit, the upper triangular matrix T = Q**T B Z.  The
              elements below the diagonal are set to zero.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the orthogonal matrix Q1,
              typically from the QR factorization of B.
              On exit, if COMPQ='I', the orthogonal matrix Q, and if
              COMPQ = 'V', the product Q1*Q.
              Not referenced if COMPQ='N'.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the orthogonal matrix Z1.
              On exit, if COMPZ='I', the orthogonal matrix Z, and if
              COMPZ = 'V', the product Z1*Z.
              Not referenced if COMPZ='N'.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.
              LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      This routine reduces A to Hessenberg and B to triangular form by
      an unblocked reduction, as described in _Matrix_Computations_,
      by Golub and Van Loan (Johns Hopkins Press.)

Definition at line 205 of file dgghrd.f.

  subroutine dggqrf (integer n, integer m, integer p, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) taua, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) taub, double precision, dimension( * ) work, integer lwork, integer info)

DGGQRF

Purpose:

     DGGQRF computes a generalized QR factorization of an N-by-M matrix A
     and an N-by-P matrix B:
                 A = Q*R,        B = Q*T*Z,
     where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
     matrix, and R and T assume one of the forms:
     if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                     (  0  ) N-M                         N   M-N
                        M
     where R11 is upper triangular, and
     if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                      P-N  N                           ( T21 ) P
                                                          P
     where T12 or T21 is upper triangular.
     In particular, if B is square and nonsingular, the GQR factorization
     of A and B implicitly gives the QR factorization of inv(B)*A:
                  inv(B)*A = Z**T*(inv(T)*R)
     where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
     transpose of the matrix Z.

Parameters
    N

              N is INTEGER
              The number of rows of the matrices A and B. N >= 0.

    M

              M is INTEGER
              The number of columns of the matrix A.  M >= 0.

    P

              P is INTEGER
              The number of columns of the matrix B.  P >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,M)
              On entry, the N-by-M matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(N,M)-by-M upper trapezoidal matrix R (R is
              upper triangular if N >= M); the elements below the diagonal,
              with the array TAUA, represent the orthogonal matrix Q as a
              product of min(N,M) elementary reflectors (see Further
              Details).

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,N).

    TAUA

              TAUA is DOUBLE PRECISION array, dimension (min(N,M))
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Q (see Further Details).

    B

              B is DOUBLE PRECISION array, dimension (LDB,P)
              On entry, the N-by-P matrix B.
              On exit, if N <= P, the upper triangle of the subarray
              B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
              if N > P, the elements on and above the (N-P)-th subdiagonal
              contain the N-by-P upper trapezoidal matrix T; the remaining
              elements, with the array TAUB, represent the orthogonal
              matrix Z as a product of elementary reflectors (see Further
              Details).

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,N).

    TAUB

              TAUB is DOUBLE PRECISION array, dimension (min(N,P))
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Z (see Further Details).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N,M,P).
              For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
              where NB1 is the optimal blocksize for the QR factorization
              of an N-by-M matrix, NB2 is the optimal blocksize for the
              RQ factorization of an N-by-P matrix, and NB3 is the optimal
              blocksize for a call of DORMQR.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(1) H(2) . . . H(k), where k = min(n,m).
      Each H(i) has the form
         H(i) = I - taua * v * v**T
      where taua is a real scalar, and v is a real vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
      and taua in TAUA(i).
      To form Q explicitly, use LAPACK subroutine DORGQR.
      To use Q to update another matrix, use LAPACK subroutine DORMQR.
      The matrix Z is represented as a product of elementary reflectors
         Z = H(1) H(2) . . . H(k), where k = min(n,p).
      Each H(i) has the form
         H(i) = I - taub * v * v**T
      where taub is a real scalar, and v is a real vector with
      v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
      B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
      To form Z explicitly, use LAPACK subroutine DORGRQ.
      To use Z to update another matrix, use LAPACK subroutine DORMRQ.

Definition at line 213 of file dggqrf.f.

  subroutine dggrqf (integer m, integer p, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) taua, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) taub, double precision, dimension( * ) work, integer lwork, integer info)

DGGRQF

Purpose:

     DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
     and a P-by-N matrix B:
                 A = R*Q,        B = Z*T*Q,
     where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
     matrix, and R and T assume one of the forms:
     if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                      N-M  M                           ( R21 ) N
                                                          N
     where R12 or R21 is upper triangular, and
     if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
                     (  0  ) P-N                         P   N-P
                        N
     where T11 is upper triangular.
     In particular, if B is square and nonsingular, the GRQ factorization
     of A and B implicitly gives the RQ factorization of A*inv(B):
                  A*inv(B) = (R*inv(T))*Z**T
     where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
     transpose of the matrix Z.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    P

              P is INTEGER
              The number of rows of the matrix B.  P >= 0.

    N

              N is INTEGER
              The number of columns of the matrices A and B. N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, if M <= N, the upper triangle of the subarray
              A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
              if M > N, the elements on and above the (M-N)-th subdiagonal
              contain the M-by-N upper trapezoidal matrix R; the remaining
              elements, with the array TAUA, represent the orthogonal
              matrix Q as a product of elementary reflectors (see Further
              Details).

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    TAUA

              TAUA is DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Q (see Further Details).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, the elements on and above the diagonal of the array
              contain the min(P,N)-by-N upper trapezoidal matrix T (T is
              upper triangular if P >= N); the elements below the diagonal,
              with the array TAUB, represent the orthogonal matrix Z as a
              product of elementary reflectors (see Further Details).

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,P).

    TAUB

              TAUB is DOUBLE PRECISION array, dimension (min(P,N))
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Z (see Further Details).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N,M,P).
              For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
              where NB1 is the optimal blocksize for the RQ factorization
              of an M-by-N matrix, NB2 is the optimal blocksize for the
              QR factorization of a P-by-N matrix, and NB3 is the optimal
              blocksize for a call of DORMRQ.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INF0= -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(1) H(2) . . . H(k), where k = min(m,n).
      Each H(i) has the form
         H(i) = I - taua * v * v**T
      where taua is a real scalar, and v is a real vector with
      v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
      A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
      To form Q explicitly, use LAPACK subroutine DORGRQ.
      To use Q to update another matrix, use LAPACK subroutine DORMRQ.
      The matrix Z is represented as a product of elementary reflectors
         Z = H(1) H(2) . . . H(k), where k = min(p,n).
      Each H(i) has the form
         H(i) = I - taub * v * v**T
      where taub is a real scalar, and v is a real vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
      and taub in TAUB(i).
      To form Z explicitly, use LAPACK subroutine DORGQR.
      To use Z to update another matrix, use LAPACK subroutine DORMQR.

Definition at line 212 of file dggrqf.f.

  subroutine dggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, integer k, integer l, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DGGSVP3

Purpose:

     DGGSVP3 computes orthogonal matrices U, V and Q such that
                        N-K-L  K    L
      U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
                     L ( 0     0   A23 )
                 M-K-L ( 0     0    0  )
                      N-K-L  K    L
             =     K ( 0    A12  A13 )  if M-K-L < 0;
                 M-K ( 0     0   A23 )
                      N-K-L  K    L
      V**T*B*Q =   L ( 0     0   B13 )
                 P-L ( 0     0    0  )
     where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
     upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
     otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
     numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
     This decomposition is the preprocessing step for computing the
     Generalized Singular Value Decomposition (GSVD), see subroutine
     DGGSVD3.

Parameters
    JOBU

              JOBU is CHARACTER*1
              = 'U':  Orthogonal matrix U is computed;
              = 'N':  U is not computed.

    JOBV

              JOBV is CHARACTER*1
              = 'V':  Orthogonal matrix V is computed;
              = 'N':  V is not computed.

    JOBQ

              JOBQ is CHARACTER*1
              = 'Q':  Orthogonal matrix Q is computed;
              = 'N':  Q is not computed.

    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    P

              P is INTEGER
              The number of rows of the matrix B.  P >= 0.

    N

              N is INTEGER
              The number of columns of the matrices A and B.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A contains the triangular (or trapezoidal) matrix
              described in the Purpose section.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, B contains the triangular matrix described in
              the Purpose section.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,P).

    TOLA

              TOLA is DOUBLE PRECISION

    TOLB

              TOLB is DOUBLE PRECISION
              TOLA and TOLB are the thresholds to determine the effective
              numerical rank of matrix B and a subblock of A. Generally,
              they are set to
                 TOLA = MAX(M,N)*norm(A)*MACHEPS,
                 TOLB = MAX(P,N)*norm(B)*MACHEPS.
              The size of TOLA and TOLB may affect the size of backward
              errors of the decomposition.

    K

              K is INTEGER

    L

              L is INTEGER
              On exit, K and L specify the dimension of the subblocks
              described in Purpose section.
              K + L = effective numerical rank of (A**T,B**T)**T.

    U

              U is DOUBLE PRECISION array, dimension (LDU,M)
              If JOBU = 'U', U contains the orthogonal matrix U.
              If JOBU = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U. LDU >= max(1,M) if
              JOBU = 'U'; LDU >= 1 otherwise.

    V

              V is DOUBLE PRECISION array, dimension (LDV,P)
              If JOBV = 'V', V contains the orthogonal matrix V.
              If JOBV = 'N', V is not referenced.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,P) if
              JOBV = 'V'; LDV >= 1 otherwise.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              If JOBQ = 'Q', Q contains the orthogonal matrix Q.
              If JOBQ = 'N', Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q. LDQ >= max(1,N) if
              JOBQ = 'Q'; LDQ >= 1 otherwise.

    IWORK

              IWORK is INTEGER array, dimension (N)

    TAU

              TAU is DOUBLE PRECISION array, dimension (N)

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The subroutine uses LAPACK subroutine DGEQP3 for the QR factorization
      with column pivoting to detect the effective numerical rank of the
      a matrix. It may be replaced by a better rank determination strategy.
      DGGSVP3 replaces the deprecated subroutine DGGSVP.

Definition at line 269 of file dggsvp3.f.

  subroutine dgsvj0 (character*1 jobv, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( n ) d, double precision, dimension( n ) sva, integer mv, double precision, dimension( ldv, * ) v, integer ldv, double precision eps, double precision sfmin, double precision tol, integer nsweep, double precision, dimension( lwork ) work, integer lwork, integer info)

DGSVJ0 pre-processor for the routine dgesvj.

Purpose:

     DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
     purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
     it does not check convergence (stopping criterion). Few tuning
     parameters (marked by [TP]) are available for the implementer.

Parameters
    JOBV

              JOBV is CHARACTER*1
              Specifies whether the output from this procedure is used
              to compute the matrix V:
              = 'V': the product of the Jacobi rotations is accumulated
                     by postmulyiplying the N-by-N array V.
                    (See the description of V.)
              = 'A': the product of the Jacobi rotations is accumulated
                     by postmulyiplying the MV-by-N array V.
                    (See the descriptions of MV and V.)
              = 'N': the Jacobi rotations are not accumulated.

    M

              M is INTEGER
              The number of rows of the input matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the input matrix A.
              M >= N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, M-by-N matrix A, such that A*diag(D) represents
              the input matrix.
              On exit,
              A_onexit * D_onexit represents the input matrix A*diag(D)
              post-multiplied by a sequence of Jacobi rotations, where the
              rotation threshold and the total number of sweeps are given in
              TOL and NSWEEP, respectively.
              (See the descriptions of D, TOL and NSWEEP.)

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    D

              D is DOUBLE PRECISION array, dimension (N)
              The array D accumulates the scaling factors from the fast scaled
              Jacobi rotations.
              On entry, A*diag(D) represents the input matrix.
              On exit, A_onexit*diag(D_onexit) represents the input matrix
              post-multiplied by a sequence of Jacobi rotations, where the
              rotation threshold and the total number of sweeps are given in
              TOL and NSWEEP, respectively.
              (See the descriptions of A, TOL and NSWEEP.)

    SVA

              SVA is DOUBLE PRECISION array, dimension (N)
              On entry, SVA contains the Euclidean norms of the columns of
              the matrix A*diag(D).
              On exit, SVA contains the Euclidean norms of the columns of
              the matrix onexit*diag(D_onexit).

    MV

              MV is INTEGER
              If JOBV = 'A', then MV rows of V are post-multipled by a
                               sequence of Jacobi rotations.
              If JOBV = 'N',   then MV is not referenced.

    V

              V is DOUBLE PRECISION array, dimension (LDV,N)
              If JOBV = 'V' then N rows of V are post-multipled by a
                               sequence of Jacobi rotations.
              If JOBV = 'A' then MV rows of V are post-multipled by a
                               sequence of Jacobi rotations.
              If JOBV = 'N',   then V is not referenced.

    LDV

              LDV is INTEGER
              The leading dimension of the array V,  LDV >= 1.
              If JOBV = 'V', LDV >= N.
              If JOBV = 'A', LDV >= MV.

    EPS

              EPS is DOUBLE PRECISION
              EPS = DLAMCH('Epsilon')

    SFMIN

              SFMIN is DOUBLE PRECISION
              SFMIN = DLAMCH('Safe Minimum')

    TOL

              TOL is DOUBLE PRECISION
              TOL is the threshold for Jacobi rotations. For a pair
              A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
              applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.

    NSWEEP

              NSWEEP is INTEGER
              NSWEEP is the number of sweeps of Jacobi rotations to be
              performed.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              LWORK is the dimension of WORK. LWORK >= M.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, then the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:
    DGSVJ0 is used just to enable DGESVJ to call a simplified version of itself to work on a submatrix of the original matrix.

Contributors:
    Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

Bugs, Examples and Comments:
    Please report all bugs and send interesting test examples and comments to drmac@math.hr. Thank you.

Definition at line 216 of file dgsvj0.f.

  subroutine dgsvj1 (character*1 jobv, integer m, integer n, integer n1, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( n ) d, double precision, dimension( n ) sva, integer mv, double precision, dimension( ldv, * ) v, integer ldv, double precision eps, double precision sfmin, double precision tol, integer nsweep, double precision, dimension( lwork ) work, integer lwork, integer info)

DGSVJ1 pre-processor for the routine dgesvj, applies Jacobi rotations targeting only particular pivots.

Purpose:

     DGSVJ1 is called from DGESVJ as a pre-processor and that is its main
     purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
     it targets only particular pivots and it does not check convergence
     (stopping criterion). Few tuning parameters (marked by [TP]) are
     available for the implementer.
     Further Details
     ~~~~~~~~~~~~~~~
     DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
     the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
     off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
     block-entries (tiles) of the (1,2) off-diagonal block are marked by the
     [x]'s in the following scheme:
        | *  *  * [x] [x] [x]|
        | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
        | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
        |[x] [x] [x] *  *  * |
        |[x] [x] [x] *  *  * |
        |[x] [x] [x] *  *  * |
     In terms of the columns of A, the first N1 columns are rotated 'against'
     the remaining N-N1 columns, trying to increase the angle between the
     corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
     tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
     The number of sweeps is given in NSWEEP and the orthogonality threshold
     is given in TOL.

Parameters
    JOBV

              JOBV is CHARACTER*1
              Specifies whether the output from this procedure is used
              to compute the matrix V:
              = 'V': the product of the Jacobi rotations is accumulated
                     by postmulyiplying the N-by-N array V.
                    (See the description of V.)
              = 'A': the product of the Jacobi rotations is accumulated
                     by postmulyiplying the MV-by-N array V.
                    (See the descriptions of MV and V.)
              = 'N': the Jacobi rotations are not accumulated.

    M

              M is INTEGER
              The number of rows of the input matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the input matrix A.
              M >= N >= 0.

    N1

              N1 is INTEGER
              N1 specifies the 2 x 2 block partition, the first N1 columns are
              rotated 'against' the remaining N-N1 columns of A.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, M-by-N matrix A, such that A*diag(D) represents
              the input matrix.
              On exit,
              A_onexit * D_onexit represents the input matrix A*diag(D)
              post-multiplied by a sequence of Jacobi rotations, where the
              rotation threshold and the total number of sweeps are given in
              TOL and NSWEEP, respectively.
              (See the descriptions of N1, D, TOL and NSWEEP.)

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    D

              D is DOUBLE PRECISION array, dimension (N)
              The array D accumulates the scaling factors from the fast scaled
              Jacobi rotations.
              On entry, A*diag(D) represents the input matrix.
              On exit, A_onexit*diag(D_onexit) represents the input matrix
              post-multiplied by a sequence of Jacobi rotations, where the
              rotation threshold and the total number of sweeps are given in
              TOL and NSWEEP, respectively.
              (See the descriptions of N1, A, TOL and NSWEEP.)

    SVA

              SVA is DOUBLE PRECISION array, dimension (N)
              On entry, SVA contains the Euclidean norms of the columns of
              the matrix A*diag(D).
              On exit, SVA contains the Euclidean norms of the columns of
              the matrix onexit*diag(D_onexit).

    MV

              MV is INTEGER
              If JOBV = 'A', then MV rows of V are post-multipled by a
                             sequence of Jacobi rotations.
              If JOBV = 'N', then MV is not referenced.

    V

              V is DOUBLE PRECISION array, dimension (LDV,N)
              If JOBV = 'V', then N rows of V are post-multipled by a
                             sequence of Jacobi rotations.
              If JOBV = 'A', then MV rows of V are post-multipled by a
                             sequence of Jacobi rotations.
              If JOBV = 'N', then V is not referenced.

    LDV

              LDV is INTEGER
              The leading dimension of the array V,  LDV >= 1.
              If JOBV = 'V', LDV >= N.
              If JOBV = 'A', LDV >= MV.

    EPS

              EPS is DOUBLE PRECISION
              EPS = DLAMCH('Epsilon')

    SFMIN

              SFMIN is DOUBLE PRECISION
              SFMIN = DLAMCH('Safe Minimum')

    TOL

              TOL is DOUBLE PRECISION
              TOL is the threshold for Jacobi rotations. For a pair
              A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
              applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.

    NSWEEP

              NSWEEP is INTEGER
              NSWEEP is the number of sweeps of Jacobi rotations to be
              performed.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              LWORK is the dimension of WORK. LWORK >= M.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, then the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

Definition at line 234 of file dgsvj1.f.

  subroutine dhsein (character side, character eigsrc, character initv, logical, dimension( * ) select, integer n, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer, dimension( * ) ifaill, integer, dimension( * ) ifailr, integer info)

DHSEIN

Purpose:

     DHSEIN uses inverse iteration to find specified right and/or left
     eigenvectors of a real upper Hessenberg matrix H.
     The right eigenvector x and the left eigenvector y of the matrix H
     corresponding to an eigenvalue w are defined by:
                  H * x = w * x,     y**h * H = w * y**h
     where y**h denotes the conjugate transpose of the vector y.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'R': compute right eigenvectors only;
              = 'L': compute left eigenvectors only;
              = 'B': compute both right and left eigenvectors.

    EIGSRC

              EIGSRC is CHARACTER*1
              Specifies the source of eigenvalues supplied in (WR,WI):
              = 'Q': the eigenvalues were found using DHSEQR; thus, if
                     H has zero subdiagonal elements, and so is
                     block-triangular, then the j-th eigenvalue can be
                     assumed to be an eigenvalue of the block containing
                     the j-th row/column.  This property allows DHSEIN to
                     perform inverse iteration on just one diagonal block.
              = 'N': no assumptions are made on the correspondence
                     between eigenvalues and diagonal blocks.  In this
                     case, DHSEIN must always perform inverse iteration
                     using the whole matrix H.

    INITV

              INITV is CHARACTER*1
              = 'N': no initial vectors are supplied;
              = 'U': user-supplied initial vectors are stored in the arrays
                     VL and/or VR.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              Specifies the eigenvectors to be computed. To select the
              real eigenvector corresponding to a real eigenvalue WR(j),
              SELECT(j) must be set to .TRUE.. To select the complex
              eigenvector corresponding to a complex eigenvalue
              (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
              either SELECT(j) or SELECT(j+1) or both must be set to
              .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
              .FALSE..

    N

              N is INTEGER
              The order of the matrix H.  N >= 0.

    H

              H is DOUBLE PRECISION array, dimension (LDH,N)
              The upper Hessenberg matrix H.
              If a NaN is detected in H, the routine will return with INFO=-6.

    LDH

              LDH is INTEGER
              The leading dimension of the array H.  LDH >= max(1,N).

    WR

              WR is DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION array, dimension (N)
              On entry, the real and imaginary parts of the eigenvalues of
              H; a complex conjugate pair of eigenvalues must be stored in
              consecutive elements of WR and WI.
              On exit, WR may have been altered since close eigenvalues
              are perturbed slightly in searching for independent
              eigenvectors.

    VL

              VL is DOUBLE PRECISION array, dimension (LDVL,MM)
              On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
              contain starting vectors for the inverse iteration for the
              left eigenvectors; the starting vector for each eigenvector
              must be in the same column(s) in which the eigenvector will
              be stored.
              On exit, if SIDE = 'L' or 'B', the left eigenvectors
              specified by SELECT will be stored consecutively in the
              columns of VL, in the same order as their eigenvalues. A
              complex eigenvector corresponding to a complex eigenvalue is
              stored in two consecutive columns, the first holding the real
              part and the second the imaginary part.
              If SIDE = 'R', VL is not referenced.

    LDVL

              LDVL is INTEGER
              The leading dimension of the array VL.
              LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,MM)
              On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
              contain starting vectors for the inverse iteration for the
              right eigenvectors; the starting vector for each eigenvector
              must be in the same column(s) in which the eigenvector will
              be stored.
              On exit, if SIDE = 'R' or 'B', the right eigenvectors
              specified by SELECT will be stored consecutively in the
              columns of VR, in the same order as their eigenvalues. A
              complex eigenvector corresponding to a complex eigenvalue is
              stored in two consecutive columns, the first holding the real
              part and the second the imaginary part.
              If SIDE = 'L', VR is not referenced.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR.
              LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

    MM

              MM is INTEGER
              The number of columns in the arrays VL and/or VR. MM >= M.

    M

              M is INTEGER
              The number of columns in the arrays VL and/or VR required to
              store the eigenvectors; each selected real eigenvector
              occupies one column and each selected complex eigenvector
              occupies two columns.

    WORK

              WORK is DOUBLE PRECISION array, dimension ((N+2)*N)

    IFAILL

              IFAILL is INTEGER array, dimension (MM)
              If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
              eigenvector in the i-th column of VL (corresponding to the
              eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
              eigenvector converged satisfactorily. If the i-th and (i+1)th
              columns of VL hold a complex eigenvector, then IFAILL(i) and
              IFAILL(i+1) are set to the same value.
              If SIDE = 'R', IFAILL is not referenced.

    IFAILR

              IFAILR is INTEGER array, dimension (MM)
              If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
              eigenvector in the i-th column of VR (corresponding to the
              eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
              eigenvector converged satisfactorily. If the i-th and (i+1)th
              columns of VR hold a complex eigenvector, then IFAILR(i) and
              IFAILR(i+1) are set to the same value.
              If SIDE = 'L', IFAILR is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, i is the number of eigenvectors which
                    failed to converge; see IFAILL and IFAILR for further
                    details.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Each eigenvector is normalized so that the element of largest
      magnitude has magnitude 1; here the magnitude of a complex number
      (x,y) is taken to be |x|+|y|.

Definition at line 260 of file dhsein.f.

  subroutine dhseqr (character job, character compz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)

DHSEQR

Purpose:

        DHSEQR computes the eigenvalues of a Hessenberg matrix H
        and, optionally, the matrices T and Z from the Schur decomposition
        H = Z T Z**T, where T is an upper quasi-triangular matrix (the
        Schur form), and Z is the orthogonal matrix of Schur vectors.
        Optionally Z may be postmultiplied into an input orthogonal
        matrix Q so that this routine can give the Schur factorization
        of a matrix A which has been reduced to the Hessenberg form H
        by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

Parameters
    JOB

              JOB is CHARACTER*1
               = 'E':  compute eigenvalues only;
               = 'S':  compute eigenvalues and the Schur form T.

    COMPZ

              COMPZ is CHARACTER*1
               = 'N':  no Schur vectors are computed;
               = 'I':  Z is initialized to the unit matrix and the matrix Z
                       of Schur vectors of H is returned;
               = 'V':  Z must contain an orthogonal matrix Q on entry, and
                       the product Q*Z is returned.

    N

              N is INTEGER
               The order of the matrix H.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
               It is assumed that H is already upper triangular in rows
               and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
               set by a previous call to DGEBAL, and then passed to ZGEHRD
               when the matrix output by DGEBAL is reduced to Hessenberg
               form. Otherwise ILO and IHI should be set to 1 and N
               respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
               If N = 0, then ILO = 1 and IHI = 0.

    H

              H is DOUBLE PRECISION array, dimension (LDH,N)
               On entry, the upper Hessenberg matrix H.
               On exit, if INFO = 0 and JOB = 'S', then H contains the
               upper quasi-triangular matrix T from the Schur decomposition
               (the Schur form); 2-by-2 diagonal blocks (corresponding to
               complex conjugate pairs of eigenvalues) are returned in
               standard form, with H(i,i) = H(i+1,i+1) and
               H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
               contents of H are unspecified on exit.  (The output value of
               H when INFO > 0 is given under the description of INFO
               below.)
               Unlike earlier versions of DHSEQR, this subroutine may
               explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
               or j = IHI+1, IHI+2, ... N.

    LDH

              LDH is INTEGER
               The leading dimension of the array H. LDH >= max(1,N).

    WR

              WR is DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION array, dimension (N)
               The real and imaginary parts, respectively, of the computed
               eigenvalues. If two eigenvalues are computed as a complex
               conjugate pair, they are stored in consecutive elements of
               WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and
               WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in
               the same order as on the diagonal of the Schur form returned
               in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
               diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
               WI(i+1) = -WI(i).

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ,N)
               If COMPZ = 'N', Z is not referenced.
               If COMPZ = 'I', on entry Z need not be set and on exit,
               if INFO = 0, Z contains the orthogonal matrix Z of the Schur
               vectors of H.  If COMPZ = 'V', on entry Z must contain an
               N-by-N matrix Q, which is assumed to be equal to the unit
               matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
               if INFO = 0, Z contains Q*Z.
               Normally Q is the orthogonal matrix generated by DORGHR
               after the call to DGEHRD which formed the Hessenberg matrix
               H. (The output value of Z when INFO > 0 is given under
               the description of INFO below.)

    LDZ

              LDZ is INTEGER
               The leading dimension of the array Z.  if COMPZ = 'I' or
               COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
               On exit, if INFO = 0, WORK(1) returns an estimate of
               the optimal value for LWORK.

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK.  LWORK >= max(1,N)
               is sufficient and delivers very good and sometimes
               optimal performance.  However, LWORK as large as 11*N
               may be required for optimal performance.  A workspace
               query is recommended to determine the optimal workspace
               size.
               If LWORK = -1, then DHSEQR does a workspace query.
               In this case, DHSEQR checks the input parameters and
               estimates the optimal workspace size for the given
               values of N, ILO and IHI.  The estimate is returned
               in WORK(1).  No error message related to LWORK is
               issued by XERBLA.  Neither H nor Z are accessed.

    INFO

              INFO is INTEGER
                 = 0:  successful exit
                 < 0:  if INFO = -i, the i-th argument had an illegal
                        value
                 > 0:  if INFO = i, DHSEQR failed to compute all of
                    the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                    and WI contain those eigenvalues which have been
                    successfully computed.  (Failures are rare.)
                    If INFO > 0 and JOB = 'E', then on exit, the
                    remaining unconverged eigenvalues are the eigen-
                    values of the upper Hessenberg matrix rows and
                    columns ILO through INFO of the final, output
                    value of H.
                    If INFO > 0 and JOB   = 'S', then on exit
               (*)  (initial value of H)*U  = U*(final value of H)
                    where U is an orthogonal matrix.  The final
                    value of H is upper Hessenberg and quasi-triangular
                    in rows and columns INFO+1 through IHI.
                    If INFO > 0 and COMPZ = 'V', then on exit
                      (final value of Z)  =  (initial value of Z)*U
                    where U is the orthogonal matrix in (*) (regard-
                    less of the value of JOB.)
                    If INFO > 0 and COMPZ = 'I', then on exit
                          (final value of Z)  = U
                    where U is the orthogonal matrix in (*) (regard-
                    less of the value of JOB.)
                    If INFO > 0 and COMPZ = 'N', then Z is not
                    accessed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Further Details:

                 Default values supplied by
                 ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
                 It is suggested that these defaults be adjusted in order
                 to attain best performance in each particular
                 computational environment.
                ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
                          Default: 75. (Must be at least 11.)
                ISPEC=13: Recommended deflation window size.
                          This depends on ILO, IHI and NS.  NS is the
                          number of simultaneous shifts returned
                          by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
                          The default for (IHI-ILO+1) <= 500 is NS.
                          The default for (IHI-ILO+1) >  500 is 3*NS/2.
                ISPEC=14: Nibble crossover point. (See IPARMQ for
                          details.)  Default: 14% of deflation window
                          size.
                ISPEC=15: Number of simultaneous shifts in a multishift
                          QR iteration.
                          If IHI-ILO+1 is ...
                          greater than      ...but less    ... the
                          or equal to ...      than        default is
                               1               30          NS =   2(+)
                              30               60          NS =   4(+)
                              60              150          NS =  10(+)
                             150              590          NS =  **
                             590             3000          NS =  64
                            3000             6000          NS = 128
                            6000             infinity      NS = 256
                      (+)  By default some or all matrices of this order
                           are passed to the implicit double shift routine
                           DLAHQR and this parameter is ignored.  See
                           ISPEC=12 above and comments in IPARMQ for
                           details.
                     (**)  The asterisks (**) indicate an ad-hoc
                           function of N increasing from 10 to 64.
                ISPEC=16: Select structured matrix multiply.
                          If the number of simultaneous shifts (specified
                          by ISPEC=15) is less than 14, then the default
                          for ISPEC=16 is 0.  Otherwise the default for
                          ISPEC=16 is 2.

References:

      K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
      Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
      Performance, SIAM Journal of Matrix Analysis, volume 23, pages
      929--947, 2002.

     K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 314 of file dhseqr.f.

  subroutine dla_lin_berr (integer n, integer nz, integer nrhs, double precision, dimension( n, nrhs ) res, double precision, dimension( n, nrhs ) ayb, double precision, dimension( nrhs ) berr)

DLA_LIN_BERR computes a component-wise relative backward error.

Purpose:

        DLA_LIN_BERR computes component-wise relative backward error from
        the formula
            max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
        where abs(Z) is the component-wise absolute value of the matrix
        or vector Z.

Parameters
    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    NZ

              NZ is INTEGER
         We add (NZ+1)*SLAMCH( 'Safe minimum' ) to R(i) in the numerator to
         guard against spuriously zero residuals. Default value is N.

    NRHS

              NRHS is INTEGER
         The number of right hand sides, i.e., the number of columns
         of the matrices AYB, RES, and BERR.  NRHS >= 0.

    RES

              RES is DOUBLE PRECISION array, dimension (N,NRHS)
         The residual matrix, i.e., the matrix R in the relative backward
         error formula above.

    AYB

              AYB is DOUBLE PRECISION array, dimension (N, NRHS)
         The denominator in the relative backward error formula above, i.e.,
         the matrix abs(op(A_s))*abs(Y) + abs(B_s). The matrices A, Y, and B
         are from iterative refinement (see dla_gerfsx_extended.f).

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
         The component-wise relative backward error from the formula above.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 100 of file dla_lin_berr.f.

  subroutine dla_wwaddw (integer n, double precision, dimension( * ) x, double precision, dimension( * ) y, double precision, dimension( * ) w)

DLA_WWADDW adds a vector into a doubled-single vector.

Purpose:

        DLA_WWADDW adds a vector W into a doubled-single vector (X, Y).
        This works for all extant IBM's hex and binary floating point
        arithmetic, but not for decimal.

Parameters
    N

              N is INTEGER
                The length of vectors X, Y, and W.

    X

              X is DOUBLE PRECISION array, dimension (N)
                The first part of the doubled-single accumulation vector.

    Y

              Y is DOUBLE PRECISION array, dimension (N)
                The second part of the doubled-single accumulation vector.

    W

              W is DOUBLE PRECISION array, dimension (N)
                The vector to be added.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 80 of file dla_wwaddw.f.

  subroutine dlals0 (integer icompq, integer nl, integer nr, integer sqre, integer nrhs, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldbx, * ) bx, integer ldbx, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, double precision, dimension( ldgnum, * ) givnum, integer ldgnum, double precision, dimension( ldgnum, * ) poles, double precision, dimension( * ) difl, double precision, dimension( ldgnum, * ) difr, double precision, dimension( * ) z, integer k, double precision c, double precision s, double precision, dimension( * ) work, integer info)

DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

     DLALS0 applies back the multiplying factors of either the left or the
     right singular vector matrix of a diagonal matrix appended by a row
     to the right hand side matrix B in solving the least squares problem
     using the divide-and-conquer SVD approach.
     For the left singular vector matrix, three types of orthogonal
     matrices are involved:
     (1L) Givens rotations: the number of such rotations is GIVPTR; the
          pairs of columns/rows they were applied to are stored in GIVCOL;
          and the C- and S-values of these rotations are stored in GIVNUM.
     (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
          row, and for J=2:N, PERM(J)-th row of B is to be moved to the
          J-th row.
     (3L) The left singular vector matrix of the remaining matrix.
     For the right singular vector matrix, four types of orthogonal
     matrices are involved:
     (1R) The right singular vector matrix of the remaining matrix.
     (2R) If SQRE = 1, one extra Givens rotation to generate the right
          null space.
     (3R) The inverse transformation of (2L).
     (4R) The inverse transformation of (1L).

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
             Specifies whether singular vectors are to be computed in
             factored form:
             = 0: Left singular vector matrix.
             = 1: Right singular vector matrix.

    NL

              NL is INTEGER
             The row dimension of the upper block. NL >= 1.

    NR

              NR is INTEGER
             The row dimension of the lower block. NR >= 1.

    SQRE

              SQRE is INTEGER
             = 0: the lower block is an NR-by-NR square matrix.
             = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
             The bidiagonal matrix has row dimension N = NL + NR + 1,
             and column dimension M = N + SQRE.

    NRHS

              NRHS is INTEGER
             The number of columns of B and BX. NRHS must be at least 1.

    B

              B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
             On input, B contains the right hand sides of the least
             squares problem in rows 1 through M. On output, B contains
             the solution X in rows 1 through N.

    LDB

              LDB is INTEGER
             The leading dimension of B. LDB must be at least
             max(1,MAX( M, N ) ).

    BX

              BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )

    LDBX

              LDBX is INTEGER
             The leading dimension of BX.

    PERM

              PERM is INTEGER array, dimension ( N )
             The permutations (from deflation and sorting) applied
             to the two blocks.

    GIVPTR

              GIVPTR is INTEGER
             The number of Givens rotations which took place in this
             subproblem.

    GIVCOL

              GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
             Each pair of numbers indicates a pair of rows/columns
             involved in a Givens rotation.

    LDGCOL

              LDGCOL is INTEGER
             The leading dimension of GIVCOL, must be at least N.

    GIVNUM

              GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
             Each number indicates the C or S value used in the
             corresponding Givens rotation.

    LDGNUM

              LDGNUM is INTEGER
             The leading dimension of arrays DIFR, POLES and
             GIVNUM, must be at least K.

    POLES

              POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
             On entry, POLES(1:K, 1) contains the new singular
             values obtained from solving the secular equation, and
             POLES(1:K, 2) is an array containing the poles in the secular
             equation.

    DIFL

              DIFL is DOUBLE PRECISION array, dimension ( K ).
             On entry, DIFL(I) is the distance between I-th updated
             (undeflated) singular value and the I-th (undeflated) old
             singular value.

    DIFR

              DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
             On entry, DIFR(I, 1) contains the distances between I-th
             updated (undeflated) singular value and the I+1-th
             (undeflated) old singular value. And DIFR(I, 2) is the
             normalizing factor for the I-th right singular vector.

    Z

              Z is DOUBLE PRECISION array, dimension ( K )
             Contain the components of the deflation-adjusted updating row
             vector.

    K

              K is INTEGER
             Contains the dimension of the non-deflated matrix,
             This is the order of the related secular equation. 1 <= K <=N.

    C

              C is DOUBLE PRECISION
             C contains garbage if SQRE =0 and the C-value of a Givens
             rotation related to the right null space if SQRE = 1.

    S

              S is DOUBLE PRECISION
             S contains garbage if SQRE =0 and the S-value of a Givens
             rotation related to the right null space if SQRE = 1.

    WORK

              WORK is DOUBLE PRECISION array, dimension ( K )

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 265 of file dlals0.f.

  subroutine dlalsa (integer icompq, integer smlsiz, integer n, integer nrhs, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldbx, * ) bx, integer ldbx, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldu, * ) vt, integer, dimension( * ) k, double precision, dimension( ldu, * ) difl, double precision, dimension( ldu, * ) difr, double precision, dimension( ldu, * ) z, double precision, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, double precision, dimension( ldu, * ) givnum, double precision, dimension( * ) c, double precision, dimension( * ) s, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.

Purpose:

     DLALSA is an itermediate step in solving the least squares problem
     by computing the SVD of the coefficient matrix in compact form (The
     singular vectors are computed as products of simple orthorgonal
     matrices.).
     If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
     matrix of an upper bidiagonal matrix to the right hand side; and if
     ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
     right hand side. The singular vector matrices were generated in
     compact form by DLALSA.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
             Specifies whether the left or the right singular vector
             matrix is involved.
             = 0: Left singular vector matrix
             = 1: Right singular vector matrix

    SMLSIZ

              SMLSIZ is INTEGER
             The maximum size of the subproblems at the bottom of the
             computation tree.

    N

              N is INTEGER
             The row and column dimensions of the upper bidiagonal matrix.

    NRHS

              NRHS is INTEGER
             The number of columns of B and BX. NRHS must be at least 1.

    B

              B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
             On input, B contains the right hand sides of the least
             squares problem in rows 1 through M.
             On output, B contains the solution X in rows 1 through N.

    LDB

              LDB is INTEGER
             The leading dimension of B in the calling subprogram.
             LDB must be at least max(1,MAX( M, N ) ).

    BX

              BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
             On exit, the result of applying the left or right singular
             vector matrix to B.

    LDBX

              LDBX is INTEGER
             The leading dimension of BX.

    U

              U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
             On entry, U contains the left singular vector matrices of all
             subproblems at the bottom level.

    LDU

              LDU is INTEGER, LDU = > N.
             The leading dimension of arrays U, VT, DIFL, DIFR,
             POLES, GIVNUM, and Z.

    VT

              VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
             On entry, VT**T contains the right singular vector matrices of
             all subproblems at the bottom level.

    K

              K is INTEGER array, dimension ( N ).

    DIFL

              DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
             where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

    DIFR

              DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
             On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
             distances between singular values on the I-th level and
             singular values on the (I -1)-th level, and DIFR(*, 2 * I)
             record the normalizing factors of the right singular vectors
             matrices of subproblems on I-th level.

    Z

              Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
             On entry, Z(1, I) contains the components of the deflation-
             adjusted updating row vector for subproblems on the I-th
             level.

    POLES

              POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
             On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
             singular values involved in the secular equations on the I-th
             level.

    GIVPTR

              GIVPTR is INTEGER array, dimension ( N ).
             On entry, GIVPTR( I ) records the number of Givens
             rotations performed on the I-th problem on the computation
             tree.

    GIVCOL

              GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
             On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
             locations of Givens rotations performed on the I-th level on
             the computation tree.

    LDGCOL

              LDGCOL is INTEGER, LDGCOL = > N.
             The leading dimension of arrays GIVCOL and PERM.

    PERM

              PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
             On entry, PERM(*, I) records permutations done on the I-th
             level of the computation tree.

    GIVNUM

              GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
             On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
             values of Givens rotations performed on the I-th level on the
             computation tree.

    C

              C is DOUBLE PRECISION array, dimension ( N ).
             On entry, if the I-th subproblem is not square,
             C( I ) contains the C-value of a Givens rotation related to
             the right null space of the I-th subproblem.

    S

              S is DOUBLE PRECISION array, dimension ( N ).
             On entry, if the I-th subproblem is not square,
             S( I ) contains the S-value of a Givens rotation related to
             the right null space of the I-th subproblem.

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    IWORK

              IWORK is INTEGER array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 263 of file dlalsa.f.

  subroutine dlalsd (character uplo, integer smlsiz, integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldb, * ) b, integer ldb, double precision rcond, integer rank, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLALSD uses the singular value decomposition of A to solve the least squares problem.

Purpose:

     DLALSD uses the singular value decomposition of A to solve the least
     squares problem of finding X to minimize the Euclidean norm of each
     column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
     are N-by-NRHS. The solution X overwrites B.
     The singular values of A smaller than RCOND times the largest
     singular value are treated as zero in solving the least squares
     problem; in this case a minimum norm solution is returned.
     The actual singular values are returned in D in ascending order.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    UPLO

              UPLO is CHARACTER*1
             = 'U': D and E define an upper bidiagonal matrix.
             = 'L': D and E define a  lower bidiagonal matrix.

    SMLSIZ

              SMLSIZ is INTEGER
             The maximum size of the subproblems at the bottom of the
             computation tree.

    N

              N is INTEGER
             The dimension of the  bidiagonal matrix.  N >= 0.

    NRHS

              NRHS is INTEGER
             The number of columns of B. NRHS must be at least 1.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry D contains the main diagonal of the bidiagonal
             matrix. On exit, if INFO = 0, D contains its singular values.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
             Contains the super-diagonal entries of the bidiagonal matrix.
             On exit, E has been destroyed.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
             On input, B contains the right hand sides of the least
             squares problem. On output, B contains the solution X.

    LDB

              LDB is INTEGER
             The leading dimension of B in the calling subprogram.
             LDB must be at least max(1,N).

    RCOND

              RCOND is DOUBLE PRECISION
             The singular values of A less than or equal to RCOND times
             the largest singular value are treated as zero in solving
             the least squares problem. If RCOND is negative,
             machine precision is used instead.
             For example, if diag(S)*X=B were the least squares problem,
             where diag(S) is a diagonal matrix of singular values, the
             solution would be X(i) = B(i) / S(i) if S(i) is greater than
             RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
             RCOND*max(S).

    RANK

              RANK is INTEGER
             The number of singular values of A greater than RCOND times
             the largest singular value.

    WORK

              WORK is DOUBLE PRECISION array, dimension at least
             (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
             where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

    IWORK

              IWORK is INTEGER array, dimension at least
             (3*N*NLVL + 11*N)

    INFO

              INFO is INTEGER
             = 0:  successful exit.
             < 0:  if INFO = -i, the i-th argument had an illegal value.
             > 0:  The algorithm failed to compute a singular value while
                   working on the submatrix lying in rows and columns
                   INFO/(N+1) through MOD(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 177 of file dlalsd.f.

  double precision function dlansf (character norm, character transr, character uplo, integer n, double precision, dimension( 0: * ) a, double precision, dimension( 0: * ) work)

DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.

Purpose:

     DLANSF returns the value of the one norm, or the Frobenius norm, or
     the infinity norm, or the element of largest absolute value of a
     real symmetric matrix A in RFP format.

Returns
    DLANSF

        DLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
                 (
                 ( norm1(A),         NORM = '1', 'O' or 'o'
                 (
                 ( normI(A),         NORM = 'I' or 'i'
                 (
                 ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
     where  norm1  denotes the  one norm of a matrix (maximum column sum),
     normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
     normF  denotes the  Frobenius norm of a matrix (square root of sum of
     squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies the value to be returned in DLANSF as described
              above.

    TRANSR

              TRANSR is CHARACTER*1
              Specifies whether the RFP format of A is normal or
              transposed format.
              = 'N':  RFP format is Normal;
              = 'T':  RFP format is Transpose.

    UPLO

              UPLO is CHARACTER*1
               On entry, UPLO specifies whether the RFP matrix A came from
               an upper or lower triangular matrix as follows:
               = 'U': RFP A came from an upper triangular matrix;
               = 'L': RFP A came from a lower triangular matrix.

    N

              N is INTEGER
              The order of the matrix A. N >= 0. When N = 0, DLANSF is
              set to zero.

    A

              A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
              On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
              part of the symmetric matrix A stored in RFP format. See the
              'Notes' below for more details.
              Unchanged on exit.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
              where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
              WORK is not referenced.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 208 of file dlansf.f.

  subroutine dlarscl2 (integer m, integer n, double precision, dimension( * ) d, double precision, dimension( ldx, * ) x, integer ldx)

DLARSCL2 performs reciprocal diagonal scaling on a matrix.

Purpose:

     DLARSCL2 performs a reciprocal diagonal scaling on a matrix:
       x <-- inv(D) * x
     where the diagonal matrix D is stored as a vector.
     Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
     standard.

Parameters
    M

              M is INTEGER
         The number of rows of D and X. M >= 0.

    N

              N is INTEGER
         The number of columns of X. N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (M)
         Diagonal matrix D, stored as a vector of length M.

    X

              X is DOUBLE PRECISION array, dimension (LDX,N)
         On entry, the matrix X to be scaled by D.
         On exit, the scaled matrix.

    LDX

              LDX is INTEGER
         The leading dimension of the matrix X. LDX >= M.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 89 of file dlarscl2.f.

  subroutine dlarz (character side, integer m, integer n, integer l, double precision, dimension( * ) v, integer incv, double precision tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work)

DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.

Purpose:

     DLARZ applies a real elementary reflector H to a real M-by-N
     matrix C, from either the left or the right. H is represented in the
     form
           H = I - tau * v * v**T
     where tau is a real scalar and v is a real vector.
     If tau = 0, then H is taken to be the unit matrix.
     H is a product of k elementary reflectors as returned by DTZRZF.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': form  H * C
              = 'R': form  C * H

    M

              M is INTEGER
              The number of rows of the matrix C.

    N

              N is INTEGER
              The number of columns of the matrix C.

    L

              L is INTEGER
              The number of entries of the vector V containing
              the meaningful part of the Householder vectors.
              If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

    V

              V is DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV))
              The vector v in the representation of H as returned by
              DTZRZF. V is not used if TAU = 0.

    INCV

              INCV is INTEGER
              The increment between elements of v. INCV <> 0.

    TAU

              TAU is DOUBLE PRECISION
              The value tau in the representation of H.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by the matrix H * C if SIDE = 'L',
              or C * H if SIDE = 'R'.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                             (N) if SIDE = 'L'
                          or (M) if SIDE = 'R'

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

Definition at line 144 of file dlarz.f.

  subroutine dlarzb (character side, character trans, character direct, character storev, integer m, integer n, integer k, integer l, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( ldwork, * ) work, integer ldwork)

DLARZB applies a block reflector or its transpose to a general matrix.

Purpose:

     DLARZB applies a real block reflector H or its transpose H**T to
     a real distributed M-by-N  C from the left or the right.
     Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply H or H**T from the Left
              = 'R': apply H or H**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply H (No transpose)
              = 'C': apply H**T (Transpose)

    DIRECT

              DIRECT is CHARACTER*1
              Indicates how H is formed from a product of elementary
              reflectors
              = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Indicates how the vectors which define the elementary
              reflectors are stored:
              = 'C': Columnwise                        (not supported yet)
              = 'R': Rowwise

    M

              M is INTEGER
              The number of rows of the matrix C.

    N

              N is INTEGER
              The number of columns of the matrix C.

    K

              K is INTEGER
              The order of the matrix T (= the number of elementary
              reflectors whose product defines the block reflector).

    L

              L is INTEGER
              The number of columns of the matrix V containing the
              meaningful part of the Householder reflectors.
              If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

    V

              V is DOUBLE PRECISION array, dimension (LDV,NV).
              If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.

    T

              T is DOUBLE PRECISION array, dimension (LDT,K)
              The triangular K-by-K matrix T in the representation of the
              block reflector.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (LDWORK,K)

    LDWORK

              LDWORK is INTEGER
              The leading dimension of the array WORK.
              If SIDE = 'L', LDWORK >= max(1,N);
              if SIDE = 'R', LDWORK >= max(1,M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

Definition at line 181 of file dlarzb.f.

  subroutine dlarzt (character direct, character storev, integer n, integer k, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( * ) tau, double precision, dimension( ldt, * ) t, integer ldt)

DLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:

     DLARZT forms the triangular factor T of a real block reflector
     H of order > n, which is defined as a product of k elementary
     reflectors.
     If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
     If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
     If STOREV = 'C', the vector which defines the elementary reflector
     H(i) is stored in the i-th column of the array V, and
        H  =  I - V * T * V**T
     If STOREV = 'R', the vector which defines the elementary reflector
     H(i) is stored in the i-th row of the array V, and
        H  =  I - V**T * T * V
     Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters
    DIRECT

              DIRECT is CHARACTER*1
              Specifies the order in which the elementary reflectors are
              multiplied to form the block reflector:
              = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Specifies how the vectors which define the elementary
              reflectors are stored (see also Further Details):
              = 'C': columnwise                        (not supported yet)
              = 'R': rowwise

    N

              N is INTEGER
              The order of the block reflector H. N >= 0.

    K

              K is INTEGER
              The order of the triangular factor T (= the number of
              elementary reflectors). K >= 1.

    V

              V is DOUBLE PRECISION array, dimension
                                   (LDV,K) if STOREV = 'C'
                                   (LDV,N) if STOREV = 'R'
              The matrix V. See further details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is DOUBLE PRECISION array, dimension (LDT,K)
              The k by k triangular factor T of the block reflector.
              If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
              lower triangular. The rest of the array is not used.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored; the corresponding
      array elements are modified but restored on exit. The rest of the
      array is not used.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                                  ______V_____
             ( v1 v2 v3 )                        /            \
             ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
         V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
             ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
             ( v1 v2 v3 )
                .  .  .
                .  .  .
                1  .  .
                   1  .
                      1
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                            ______V_____
                1                                          /            \
                .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
                .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
                .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
                .  .  .
             ( v1 v2 v3 )
             ( v1 v2 v3 )
         V = ( v1 v2 v3 )
             ( v1 v2 v3 )
             ( v1 v2 v3 )

Definition at line 184 of file dlarzt.f.

  subroutine dlascl2 (integer m, integer n, double precision, dimension( * ) d, double precision, dimension( ldx, * ) x, integer ldx)

DLASCL2 performs diagonal scaling on a matrix.

Purpose:

     DLASCL2 performs a diagonal scaling on a matrix:
       x <-- D * x
     where the diagonal matrix D is stored as a vector.
     Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
     standard.

Parameters
    M

              M is INTEGER
         The number of rows of D and X. M >= 0.

    N

              N is INTEGER
         The number of columns of X. N >= 0.

    D

              D is DOUBLE PRECISION array, length M
         Diagonal matrix D, stored as a vector of length M.

    X

              X is DOUBLE PRECISION array, dimension (LDX,N)
         On entry, the matrix X to be scaled by D.
         On exit, the scaled matrix.

    LDX

              LDX is INTEGER
         The leading dimension of the matrix X. LDX >= M.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 89 of file dlascl2.f.

  subroutine dlatrz (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work)

DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Purpose:

     DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
     [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
     of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
     matrix and, R and A1 are M-by-M upper triangular matrices.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A.  N >= 0.

    L

              L is INTEGER
              The number of columns of the matrix A containing the
              meaningful part of the Householder vectors. N-M >= L >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading M-by-N upper trapezoidal part of the
              array A must contain the matrix to be factorized.
              On exit, the leading M-by-M upper triangular part of A
              contains the upper triangular matrix R, and elements N-L+1 to
              N of the first M rows of A, with the array TAU, represent the
              orthogonal matrix Z as a product of M elementary reflectors.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (M)
              The scalar factors of the elementary reflectors.

    WORK

              WORK is DOUBLE PRECISION array, dimension (M)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The factorization is obtained by Householder's method.  The kth
      transformation matrix, Z( k ), which is used to introduce zeros into
      the ( m - k + 1 )th row of A, is given in the form
         Z( k ) = ( I     0   ),
                  ( 0  T( k ) )
      where
         T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
                                                     (   0    )
                                                     ( z( k ) )
      tau is a scalar and z( k ) is an l element vector. tau and z( k )
      are chosen to annihilate the elements of the kth row of A2.
      The scalar tau is returned in the kth element of TAU and the vector
      u( k ) in the kth row of A2, such that the elements of z( k ) are
      in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
      the upper triangular part of A1.
      Z is given by
         Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

Definition at line 139 of file dlatrz.f.

  subroutine dopgtr (character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) tau, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer info)

DOPGTR

Purpose:

     DOPGTR generates a real orthogonal matrix Q which is defined as the
     product of n-1 elementary reflectors H(i) of order n, as returned by
     DSPTRD using packed storage:
     if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
     if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U': Upper triangular packed storage used in previous
                     call to DSPTRD;
              = 'L': Lower triangular packed storage used in previous
                     call to DSPTRD.

    N

              N is INTEGER
              The order of the matrix Q. N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The vectors which define the elementary reflectors, as
              returned by DSPTRD.

    TAU

              TAU is DOUBLE PRECISION array, dimension (N-1)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DSPTRD.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              The N-by-N orthogonal matrix Q.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q. LDQ >= max(1,N).

    WORK

              WORK is DOUBLE PRECISION array, dimension (N-1)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 113 of file dopgtr.f.

  subroutine dopmtr (character side, character uplo, character trans, integer m, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DOPMTR

Purpose:

     DOPMTR overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix of order nq, with nq = m if
     SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
     nq-1 elementary reflectors, as returned by DSPTRD using packed
     storage:
     if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
     if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    UPLO

              UPLO is CHARACTER*1
              = 'U': Upper triangular packed storage used in previous
                     call to DSPTRD;
              = 'L': Lower triangular packed storage used in previous
                     call to DSPTRD.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension
                                   (M*(M+1)/2) if SIDE = 'L'
                                   (N*(N+1)/2) if SIDE = 'R'
              The vectors which define the elementary reflectors, as
              returned by DSPTRD.  AP is modified by the routine but
              restored on exit.

    TAU

              TAU is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L'
                                         or (N-1) if SIDE = 'R'
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DSPTRD.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                                       (N) if SIDE = 'L'
                                       (M) if SIDE = 'R'

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 148 of file dopmtr.f.

  subroutine dorbdb (character trans, character signs, integer m, integer p, integer q, double precision, dimension( ldx11, * ) x11, integer ldx11, double precision, dimension( ldx12, * ) x12, integer ldx12, double precision, dimension( ldx21, * ) x21, integer ldx21, double precision, dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double precision, dimension( * ) phi, double precision, dimension( * ) taup1, double precision, dimension( * ) taup2, double precision, dimension( * ) tauq1, double precision, dimension( * ) tauq2, double precision, dimension( * ) work, integer lwork, integer info)

DORBDB

Purpose:

     DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
     partitioned orthogonal matrix X:
                                     [ B11 | B12 0  0 ]
         [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
     X = [-----------] = [---------] [----------------] [---------]   .
         [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
                                     [  0  |  0  0  I ]
     X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
     not the case, then X must be transposed and/or permuted. This can be
     done in constant time using the TRANS and SIGNS options. See DORCSD
     for details.)
     The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
     (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
     represented implicitly by Householder vectors.
     B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
     implicitly by angles THETA, PHI.

Parameters
    TRANS

              TRANS is CHARACTER
              = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                          order;
              otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                          major order.

    SIGNS

              SIGNS is CHARACTER
              = 'O':      The lower-left block is made nonpositive (the
                          'other' convention);
              otherwise:  The upper-right block is made nonpositive (the
                          'default' convention).

    M

              M is INTEGER
              The number of rows and columns in X.

    P

              P is INTEGER
              The number of rows in X11 and X12. 0 <= P <= M.

    Q

              Q is INTEGER
              The number of columns in X11 and X21. 0 <= Q <=
              MIN(P,M-P,M-Q).

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
              On entry, the top-left block of the orthogonal matrix to be
              reduced. On exit, the form depends on TRANS:
              If TRANS = 'N', then
                 the columns of tril(X11) specify reflectors for P1,
                 the rows of triu(X11,1) specify reflectors for Q1;
              else TRANS = 'T', and
                 the rows of triu(X11) specify reflectors for P1,
                 the columns of tril(X11,-1) specify reflectors for Q1.

    LDX11

              LDX11 is INTEGER
              The leading dimension of X11. If TRANS = 'N', then LDX11 >=
              P; else LDX11 >= Q.

    X12

              X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
              On entry, the top-right block of the orthogonal matrix to
              be reduced. On exit, the form depends on TRANS:
              If TRANS = 'N', then
                 the rows of triu(X12) specify the first P reflectors for
                 Q2;
              else TRANS = 'T', and
                 the columns of tril(X12) specify the first P reflectors
                 for Q2.

    LDX12

              LDX12 is INTEGER
              The leading dimension of X12. If TRANS = 'N', then LDX12 >=
              P; else LDX11 >= M-Q.

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
              On entry, the bottom-left block of the orthogonal matrix to
              be reduced. On exit, the form depends on TRANS:
              If TRANS = 'N', then
                 the columns of tril(X21) specify reflectors for P2;
              else TRANS = 'T', and
                 the rows of triu(X21) specify reflectors for P2.

    LDX21

              LDX21 is INTEGER
              The leading dimension of X21. If TRANS = 'N', then LDX21 >=
              M-P; else LDX21 >= Q.

    X22

              X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
              On entry, the bottom-right block of the orthogonal matrix to
              be reduced. On exit, the form depends on TRANS:
              If TRANS = 'N', then
                 the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
                 M-P-Q reflectors for Q2,
              else TRANS = 'T', and
                 the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
                 M-P-Q reflectors for P2.

    LDX22

              LDX22 is INTEGER
              The leading dimension of X22. If TRANS = 'N', then LDX22 >=
              M-P; else LDX22 >= M-Q.

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
              The entries of the bidiagonal blocks B11, B12, B21, B22 can
              be computed from the angles THETA and PHI. See Further
              Details.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
              The entries of the bidiagonal blocks B11, B12, B21, B22 can
              be computed from the angles THETA and PHI. See Further
              Details.

    TAUP1

              TAUP1 is DOUBLE PRECISION array, dimension (P)
              The scalar factors of the elementary reflectors that define
              P1.

    TAUP2

              TAUP2 is DOUBLE PRECISION array, dimension (M-P)
              The scalar factors of the elementary reflectors that define
              P2.

    TAUQ1

              TAUQ1 is DOUBLE PRECISION array, dimension (Q)
              The scalar factors of the elementary reflectors that define
              Q1.

    TAUQ2

              TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
              The scalar factors of the elementary reflectors that define
              Q2.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= M-Q.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The bidiagonal blocks B11, B12, B21, and B22 are represented
      implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
      PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
      lower bidiagonal. Every entry in each bidiagonal band is a product
      of a sine or cosine of a THETA with a sine or cosine of a PHI. See
      [1] or DORCSD for details.
      P1, P2, Q1, and Q2 are represented as products of elementary
      reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
      using DORGQR and DORGLQ.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 284 of file dorbdb.f.

  subroutine dorbdb1 (integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, double precision, dimension(*) taup1, double precision, dimension(*) taup2, double precision, dimension(*) tauq1, double precision, dimension(*) work, integer lwork, integer info)

DORBDB1

Purpose:

     DORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
     matrix X with orthonomal columns:
                                [ B11 ]
          [ X11 ]   [ P1 |    ] [  0  ]
          [-----] = [---------] [-----] Q1**T .
          [ X21 ]   [    | P2 ] [ B21 ]
                                [  0  ]
     X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
     M-P, or M-Q. Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in
     which Q is not the minimum dimension.
     The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
     and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
     Householder vectors.
     B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
     angles THETA, PHI.

Parameters
    M

              M is INTEGER
               The number of rows X11 plus the number of rows in X21.

    P

              P is INTEGER
               The number of rows in X11. 0 <= P <= M.

    Q

              Q is INTEGER
               The number of columns in X11 and X21. 0 <= Q <=
               MIN(P,M-P,M-Q).

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
               On entry, the top block of the matrix X to be reduced. On
               exit, the columns of tril(X11) specify reflectors for P1 and
               the rows of triu(X11,1) specify reflectors for Q1.

    LDX11

              LDX11 is INTEGER
               The leading dimension of X11. LDX11 >= P.

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
               On entry, the bottom block of the matrix X to be reduced. On
               exit, the columns of tril(X21) specify reflectors for P2.

    LDX21

              LDX21 is INTEGER
               The leading dimension of X21. LDX21 >= M-P.

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    TAUP1

              TAUP1 is DOUBLE PRECISION array, dimension (P)
               The scalar factors of the elementary reflectors that define
               P1.

    TAUP2

              TAUP2 is DOUBLE PRECISION array, dimension (M-P)
               The scalar factors of the elementary reflectors that define
               P2.

    TAUQ1

              TAUQ1 is DOUBLE PRECISION array, dimension (Q)
               The scalar factors of the elementary reflectors that define
               Q1.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK. LWORK >= M-Q.
               If LWORK = -1, then a workspace query is assumed; the routine
               only calculates the optimal size of the WORK array, returns
               this value as the first entry of the WORK array, and no error
               message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The upper-bidiagonal blocks B11, B21 are represented implicitly by
      angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
      in each bidiagonal band is a product of a sine or cosine of a THETA
      with a sine or cosine of a PHI. See [1] or DORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
      and DORGLQ.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 201 of file dorbdb1.f.

  subroutine dorbdb2 (integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, double precision, dimension(*) taup1, double precision, dimension(*) taup2, double precision, dimension(*) tauq1, double precision, dimension(*) work, integer lwork, integer info)

DORBDB2

Purpose:

     DORBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
     matrix X with orthonomal columns:
                                [ B11 ]
          [ X11 ]   [ P1 |    ] [  0  ]
          [-----] = [---------] [-----] Q1**T .
          [ X21 ]   [    | P2 ] [ B21 ]
                                [  0  ]
     X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
     Q, or M-Q. Routines DORBDB1, DORBDB3, and DORBDB4 handle cases in
     which P is not the minimum dimension.
     The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
     and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
     Householder vectors.
     B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
     angles THETA, PHI.

Parameters
    M

              M is INTEGER
               The number of rows X11 plus the number of rows in X21.

    P

              P is INTEGER
               The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).

    Q

              Q is INTEGER
               The number of columns in X11 and X21. 0 <= Q <= M.

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
               On entry, the top block of the matrix X to be reduced. On
               exit, the columns of tril(X11) specify reflectors for P1 and
               the rows of triu(X11,1) specify reflectors for Q1.

    LDX11

              LDX11 is INTEGER
               The leading dimension of X11. LDX11 >= P.

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
               On entry, the bottom block of the matrix X to be reduced. On
               exit, the columns of tril(X21) specify reflectors for P2.

    LDX21

              LDX21 is INTEGER
               The leading dimension of X21. LDX21 >= M-P.

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    TAUP1

              TAUP1 is DOUBLE PRECISION array, dimension (P-1)
               The scalar factors of the elementary reflectors that define
               P1.

    TAUP2

              TAUP2 is DOUBLE PRECISION array, dimension (Q)
               The scalar factors of the elementary reflectors that define
               P2.

    TAUQ1

              TAUQ1 is DOUBLE PRECISION array, dimension (Q)
               The scalar factors of the elementary reflectors that define
               Q1.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK. LWORK >= M-Q.
               If LWORK = -1, then a workspace query is assumed; the routine
               only calculates the optimal size of the WORK array, returns
               this value as the first entry of the WORK array, and no error
               message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The upper-bidiagonal blocks B11, B21 are represented implicitly by
      angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
      in each bidiagonal band is a product of a sine or cosine of a THETA
      with a sine or cosine of a PHI. See [1] or DORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
      and DORGLQ.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 200 of file dorbdb2.f.

  subroutine dorbdb3 (integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, double precision, dimension(*) taup1, double precision, dimension(*) taup2, double precision, dimension(*) tauq1, double precision, dimension(*) work, integer lwork, integer info)

DORBDB3

Purpose:

     DORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
     matrix X with orthonomal columns:
                                [ B11 ]
          [ X11 ]   [ P1 |    ] [  0  ]
          [-----] = [---------] [-----] Q1**T .
          [ X21 ]   [    | P2 ] [ B21 ]
                                [  0  ]
     X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
     Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in
     which M-P is not the minimum dimension.
     The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
     and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
     Householder vectors.
     B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
     implicitly by angles THETA, PHI.

Parameters
    M

              M is INTEGER
               The number of rows X11 plus the number of rows in X21.

    P

              P is INTEGER
               The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).

    Q

              Q is INTEGER
               The number of columns in X11 and X21. 0 <= Q <= M.

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
               On entry, the top block of the matrix X to be reduced. On
               exit, the columns of tril(X11) specify reflectors for P1 and
               the rows of triu(X11,1) specify reflectors for Q1.

    LDX11

              LDX11 is INTEGER
               The leading dimension of X11. LDX11 >= P.

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
               On entry, the bottom block of the matrix X to be reduced. On
               exit, the columns of tril(X21) specify reflectors for P2.

    LDX21

              LDX21 is INTEGER
               The leading dimension of X21. LDX21 >= M-P.

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    TAUP1

              TAUP1 is DOUBLE PRECISION array, dimension (P)
               The scalar factors of the elementary reflectors that define
               P1.

    TAUP2

              TAUP2 is DOUBLE PRECISION array, dimension (M-P)
               The scalar factors of the elementary reflectors that define
               P2.

    TAUQ1

              TAUQ1 is DOUBLE PRECISION array, dimension (Q)
               The scalar factors of the elementary reflectors that define
               Q1.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK. LWORK >= M-Q.
               If LWORK = -1, then a workspace query is assumed; the routine
               only calculates the optimal size of the WORK array, returns
               this value as the first entry of the WORK array, and no error
               message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The upper-bidiagonal blocks B11, B21 are represented implicitly by
      angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
      in each bidiagonal band is a product of a sine or cosine of a THETA
      with a sine or cosine of a PHI. See [1] or DORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
      and DORGLQ.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 199 of file dorbdb3.f.

  subroutine dorbdb4 (integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, double precision, dimension(*) taup1, double precision, dimension(*) taup2, double precision, dimension(*) tauq1, double precision, dimension(*) phantom, double precision, dimension(*) work, integer lwork, integer info)

DORBDB4

Purpose:

     DORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
     matrix X with orthonomal columns:
                                [ B11 ]
          [ X11 ]   [ P1 |    ] [  0  ]
          [-----] = [---------] [-----] Q1**T .
          [ X21 ]   [    | P2 ] [ B21 ]
                                [  0  ]
     X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
     M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
     which M-Q is not the minimum dimension.
     The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
     and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
     Householder vectors.
     B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
     implicitly by angles THETA, PHI.

Parameters
    M

              M is INTEGER
               The number of rows X11 plus the number of rows in X21.

    P

              P is INTEGER
               The number of rows in X11. 0 <= P <= M.

    Q

              Q is INTEGER
               The number of columns in X11 and X21. 0 <= Q <= M and
               M-Q <= min(P,M-P,Q).

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
               On entry, the top block of the matrix X to be reduced. On
               exit, the columns of tril(X11) specify reflectors for P1 and
               the rows of triu(X11,1) specify reflectors for Q1.

    LDX11

              LDX11 is INTEGER
               The leading dimension of X11. LDX11 >= P.

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
               On entry, the bottom block of the matrix X to be reduced. On
               exit, the columns of tril(X21) specify reflectors for P2.

    LDX21

              LDX21 is INTEGER
               The leading dimension of X21. LDX21 >= M-P.

    THETA

              THETA is DOUBLE PRECISION array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

              PHI is DOUBLE PRECISION array, dimension (Q-1)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    TAUP1

              TAUP1 is DOUBLE PRECISION array, dimension (M-Q)
               The scalar factors of the elementary reflectors that define
               P1.

    TAUP2

              TAUP2 is DOUBLE PRECISION array, dimension (M-Q)
               The scalar factors of the elementary reflectors that define
               P2.

    TAUQ1

              TAUQ1 is DOUBLE PRECISION array, dimension (Q)
               The scalar factors of the elementary reflectors that define
               Q1.

    PHANTOM

              PHANTOM is DOUBLE PRECISION array, dimension (M)
               The routine computes an M-by-1 column vector Y that is
               orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
               PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
               Y(P+1:M), respectively.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK. LWORK >= M-Q.
               If LWORK = -1, then a workspace query is assumed; the routine
               only calculates the optimal size of the WORK array, returns
               this value as the first entry of the WORK array, and no error
               message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The upper-bidiagonal blocks B11, B21 are represented implicitly by
      angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
      in each bidiagonal band is a product of a sine or cosine of a THETA
      with a sine or cosine of a PHI. See [1] or DORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
      and DORGLQ.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 210 of file dorbdb4.f.

  subroutine dorbdb5 (integer m1, integer m2, integer n, double precision, dimension(*) x1, integer incx1, double precision, dimension(*) x2, integer incx2, double precision, dimension(ldq1,*) q1, integer ldq1, double precision, dimension(ldq2,*) q2, integer ldq2, double precision, dimension(*) work, integer lwork, integer info)

DORBDB5

Purpose:

     DORBDB5 orthogonalizes the column vector
          X = [ X1 ]
              [ X2 ]
     with respect to the columns of
          Q = [ Q1 ] .
              [ Q2 ]
     The columns of Q must be orthonormal.
     If the projection is zero according to Kahan's 'twice is enough'
     criterion, then some other vector from the orthogonal complement
     is returned. This vector is chosen in an arbitrary but deterministic
     way.

Parameters
    M1

              M1 is INTEGER
               The dimension of X1 and the number of rows in Q1. 0 <= M1.

    M2

              M2 is INTEGER
               The dimension of X2 and the number of rows in Q2. 0 <= M2.

    N

              N is INTEGER
               The number of columns in Q1 and Q2. 0 <= N.

    X1

              X1 is DOUBLE PRECISION array, dimension (M1)
               On entry, the top part of the vector to be orthogonalized.
               On exit, the top part of the projected vector.

    INCX1

              INCX1 is INTEGER
               Increment for entries of X1.

    X2

              X2 is DOUBLE PRECISION array, dimension (M2)
               On entry, the bottom part of the vector to be
               orthogonalized. On exit, the bottom part of the projected
               vector.

    INCX2

              INCX2 is INTEGER
               Increment for entries of X2.

    Q1

              Q1 is DOUBLE PRECISION array, dimension (LDQ1, N)
               The top part of the orthonormal basis matrix.

    LDQ1

              LDQ1 is INTEGER
               The leading dimension of Q1. LDQ1 >= M1.

    Q2

              Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
               The bottom part of the orthonormal basis matrix.

    LDQ2

              LDQ2 is INTEGER
               The leading dimension of Q2. LDQ2 >= M2.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK. LWORK >= N.

    INFO

              INFO is INTEGER
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 154 of file dorbdb5.f.

  subroutine dorbdb6 (integer m1, integer m2, integer n, double precision, dimension(*) x1, integer incx1, double precision, dimension(*) x2, integer incx2, double precision, dimension(ldq1,*) q1, integer ldq1, double precision, dimension(ldq2,*) q2, integer ldq2, double precision, dimension(*) work, integer lwork, integer info)

DORBDB6

Purpose:

     DORBDB6 orthogonalizes the column vector
          X = [ X1 ]
              [ X2 ]
     with respect to the columns of
          Q = [ Q1 ] .
              [ Q2 ]
     The Euclidean norm of X must be one and the columns of Q must be
     orthonormal. The orthogonalized vector will be zero if and only if it
     lies entirely in the range of Q.
     The projection is computed with at most two iterations of the
     classical Gram-Schmidt algorithm, see
     * L. Giraud, J. Langou, M. Rozložník. 'On the round-off error
       analysis of the Gram-Schmidt algorithm with reorthogonalization.'
       2002. CERFACS Technical Report No. TR/PA/02/33. URL:
       https://www.cerfacs.fr/algor/reports/2002/TR_PA_02_33.pdf

Parameters
    M1

              M1 is INTEGER
               The dimension of X1 and the number of rows in Q1. 0 <= M1.

    M2

              M2 is INTEGER
               The dimension of X2 and the number of rows in Q2. 0 <= M2.

    N

              N is INTEGER
               The number of columns in Q1 and Q2. 0 <= N.

    X1

              X1 is DOUBLE PRECISION array, dimension (M1)
               On entry, the top part of the vector to be orthogonalized.
               On exit, the top part of the projected vector.

    INCX1

              INCX1 is INTEGER
               Increment for entries of X1.

    X2

              X2 is DOUBLE PRECISION array, dimension (M2)
               On entry, the bottom part of the vector to be
               orthogonalized. On exit, the bottom part of the projected
               vector.

    INCX2

              INCX2 is INTEGER
               Increment for entries of X2.

    Q1

              Q1 is DOUBLE PRECISION array, dimension (LDQ1, N)
               The top part of the orthonormal basis matrix.

    LDQ1

              LDQ1 is INTEGER
               The leading dimension of Q1. LDQ1 >= M1.

    Q2

              Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
               The bottom part of the orthonormal basis matrix.

    LDQ2

              LDQ2 is INTEGER
               The leading dimension of Q2. LDQ2 >= M2.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
               The dimension of the array WORK. LWORK >= N.

    INFO

              INFO is INTEGER
               = 0:  successful exit.
               < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 158 of file dorbdb6.f.

  recursive subroutine dorcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, character signs, integer m, integer p, integer q, double precision, dimension( ldx11, * ) x11, integer ldx11, double precision, dimension( ldx12, * ) x12, integer ldx12, double precision, dimension( ldx21, * ) x21, integer ldx21, double precision, dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double precision, dimension( ldu1, * ) u1, integer ldu1, double precision, dimension( ldu2, * ) u2, integer ldu2, double precision, dimension( ldv1t, * ) v1t, integer ldv1t, double precision, dimension( ldv2t, * ) v2t, integer ldv2t, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DORCSD

Purpose:

     DORCSD computes the CS decomposition of an M-by-M partitioned
     orthogonal matrix X:
                                     [  I  0  0 |  0  0  0 ]
                                     [  0  C  0 |  0 -S  0 ]
         [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
     X = [-----------] = [---------] [---------------------] [---------]   .
         [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
                                     [  0  S  0 |  0  C  0 ]
                                     [  0  0  I |  0  0  0 ]
     X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
     (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
     R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
     which R = MIN(P,M-P,Q,M-Q).

Parameters
    JOBU1

              JOBU1 is CHARACTER
              = 'Y':      U1 is computed;
              otherwise:  U1 is not computed.

    JOBU2

              JOBU2 is CHARACTER
              = 'Y':      U2 is computed;
              otherwise:  U2 is not computed.

    JOBV1T

              JOBV1T is CHARACTER
              = 'Y':      V1T is computed;
              otherwise:  V1T is not computed.

    JOBV2T

              JOBV2T is CHARACTER
              = 'Y':      V2T is computed;
              otherwise:  V2T is not computed.

    TRANS

              TRANS is CHARACTER
              = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                          order;
              otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                          major order.

    SIGNS

              SIGNS is CHARACTER
              = 'O':      The lower-left block is made nonpositive (the
                          'other' convention);
              otherwise:  The upper-right block is made nonpositive (the
                          'default' convention).

    M

              M is INTEGER
              The number of rows and columns in X.

    P

              P is INTEGER
              The number of rows in X11 and X12. 0 <= P <= M.

    Q

              Q is INTEGER
              The number of columns in X11 and X21. 0 <= Q <= M.

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
              On entry, part of the orthogonal matrix whose CSD is desired.

    LDX11

              LDX11 is INTEGER
              The leading dimension of X11. LDX11 >= MAX(1,P).

    X12

              X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
              On entry, part of the orthogonal matrix whose CSD is desired.

    LDX12

              LDX12 is INTEGER
              The leading dimension of X12. LDX12 >= MAX(1,P).

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
              On entry, part of the orthogonal matrix whose CSD is desired.

    LDX21

              LDX21 is INTEGER
              The leading dimension of X11. LDX21 >= MAX(1,M-P).

    X22

              X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
              On entry, part of the orthogonal matrix whose CSD is desired.

    LDX22

              LDX22 is INTEGER
              The leading dimension of X11. LDX22 >= MAX(1,M-P).

    THETA

              THETA is DOUBLE PRECISION array, dimension (R), in which R =
              MIN(P,M-P,Q,M-Q).
              C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
              S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

    U1

              U1 is DOUBLE PRECISION array, dimension (LDU1,P)
              If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.

    LDU1

              LDU1 is INTEGER
              The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
              MAX(1,P).

    U2

              U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
              If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
              matrix U2.

    LDU2

              LDU2 is INTEGER
              The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
              MAX(1,M-P).

    V1T

              V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
              If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
              matrix V1**T.

    LDV1T

              LDV1T is INTEGER
              The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
              MAX(1,Q).

    V2T

              V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
              If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
              matrix V2**T.

    LDV2T

              LDV2T is INTEGER
              The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
              MAX(1,M-Q).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
              If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
              ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
              define the matrix in intermediate bidiagonal-block form
              remaining after nonconvergence. INFO specifies the number
              of nonzero PHI's.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the work array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  DBBCSD did not converge. See the description of WORK
                    above for details.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 295 of file dorcsd.f.

  subroutine dorcsd2by1 (character jobu1, character jobu2, character jobv1t, integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(ldu1,*) u1, integer ldu1, double precision, dimension(ldu2,*) u2, integer ldu2, double precision, dimension(ldv1t,*) v1t, integer ldv1t, double precision, dimension(*) work, integer lwork, integer, dimension(*) iwork, integer info)

DORCSD2BY1

Purpose:

     DORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
     orthonormal columns that has been partitioned into a 2-by-1 block
     structure:
                                    [  I1 0  0 ]
                                    [  0  C  0 ]
              [ X11 ]   [ U1 |    ] [  0  0  0 ]
          X = [-----] = [---------] [----------] V1**T .
              [ X21 ]   [    | U2 ] [  0  0  0 ]
                                    [  0  S  0 ]
                                    [  0  0  I2]
     X11 is P-by-Q. The orthogonal matrices U1, U2, and V1 are P-by-P,
     (M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R
     nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which
     R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a
     K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

Parameters
    JOBU1

              JOBU1 is CHARACTER
              = 'Y':      U1 is computed;
              otherwise:  U1 is not computed.

    JOBU2

              JOBU2 is CHARACTER
              = 'Y':      U2 is computed;
              otherwise:  U2 is not computed.

    JOBV1T

              JOBV1T is CHARACTER
              = 'Y':      V1T is computed;
              otherwise:  V1T is not computed.

    M

              M is INTEGER
              The number of rows in X.

    P

              P is INTEGER
              The number of rows in X11. 0 <= P <= M.

    Q

              Q is INTEGER
              The number of columns in X11 and X21. 0 <= Q <= M.

    X11

              X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
              On entry, part of the orthogonal matrix whose CSD is desired.

    LDX11

              LDX11 is INTEGER
              The leading dimension of X11. LDX11 >= MAX(1,P).

    X21

              X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
              On entry, part of the orthogonal matrix whose CSD is desired.

    LDX21

              LDX21 is INTEGER
              The leading dimension of X21. LDX21 >= MAX(1,M-P).

    THETA

              THETA is DOUBLE PRECISION array, dimension (R), in which R =
              MIN(P,M-P,Q,M-Q).
              C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
              S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

    U1

              U1 is DOUBLE PRECISION array, dimension (P)
              If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.

    LDU1

              LDU1 is INTEGER
              The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
              MAX(1,P).

    U2

              U2 is DOUBLE PRECISION array, dimension (M-P)
              If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
              matrix U2.

    LDU2

              LDU2 is INTEGER
              The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
              MAX(1,M-P).

    V1T

              V1T is DOUBLE PRECISION array, dimension (Q)
              If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
              matrix V1**T.

    LDV1T

              LDV1T is INTEGER
              The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
              MAX(1,Q).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
              If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
              ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
              define the matrix in intermediate bidiagonal-block form
              remaining after nonconvergence. INFO specifies the number
              of nonzero PHI's.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the work array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  DBBCSD did not converge. See the description of WORK
                    above for details.

References:
    [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 230 of file dorcsd2by1.f.

  subroutine dorg2l (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)

DORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).

Purpose:

     DORG2L generates an m by n real matrix Q with orthonormal columns,
     which is defined as the last n columns of a product of k elementary
     reflectors of order m
           Q  =  H(k) . . . H(2) H(1)
     as returned by DGEQLF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. M >= N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the (n-k+i)-th column must contain the vector which
              defines the elementary reflector H(i), for i = 1,2,...,k, as
              returned by DGEQLF in the last k columns of its array
              argument A.
              On exit, the m by n matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQLF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 113 of file dorg2l.f.

  subroutine dorg2r (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)

DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).

Purpose:

     DORG2R generates an m by n real matrix Q with orthonormal columns,
     which is defined as the first n columns of a product of k elementary
     reflectors of order m
           Q  =  H(1) H(2) . . . H(k)
     as returned by DGEQRF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. M >= N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the i-th column must contain the vector which
              defines the elementary reflector H(i), for i = 1,2,...,k, as
              returned by DGEQRF in the first k columns of its array
              argument A.
              On exit, the m-by-n matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQRF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 113 of file dorg2r.f.

  subroutine dorghr (integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DORGHR

Purpose:

     DORGHR generates a real orthogonal matrix Q which is defined as the
     product of IHI-ILO elementary reflectors of order N, as returned by
     DGEHRD:
     Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters
    N

              N is INTEGER
              The order of the matrix Q. N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI must have the same values as in the previous call
              of DGEHRD. Q is equal to the unit matrix except in the
              submatrix Q(ilo+1:ihi,ilo+1:ihi).
              1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the vectors which define the elementary reflectors,
              as returned by DGEHRD.
              On exit, the N-by-N orthogonal matrix Q.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,N).

    TAU

              TAU is DOUBLE PRECISION array, dimension (N-1)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEHRD.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= IHI-ILO.
              For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
              the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 125 of file dorghr.f.

  subroutine dorgl2 (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)

DORGL2

Purpose:

     DORGL2 generates an m by n real matrix Q with orthonormal rows,
     which is defined as the first m rows of a product of k elementary
     reflectors of order n
           Q  =  H(k) . . . H(2) H(1)
     as returned by DGELQF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. N >= M.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. M >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the i-th row must contain the vector which defines
              the elementary reflector H(i), for i = 1,2,...,k, as returned
              by DGELQF in the first k rows of its array argument A.
              On exit, the m-by-n matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGELQF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (M)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 112 of file dorgl2.f.

  subroutine dorglq (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DORGLQ

Purpose:

     DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
     which is defined as the first M rows of a product of K elementary
     reflectors of order N
           Q  =  H(k) . . . H(2) H(1)
     as returned by DGELQF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. N >= M.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. M >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the i-th row must contain the vector which defines
              the elementary reflector H(i), for i = 1,2,...,k, as returned
              by DGELQF in the first k rows of its array argument A.
              On exit, the M-by-N matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGELQF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,M).
              For optimum performance LWORK >= M*NB, where NB is
              the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 126 of file dorglq.f.

  subroutine dorgql (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DORGQL

Purpose:

     DORGQL generates an M-by-N real matrix Q with orthonormal columns,
     which is defined as the last N columns of a product of K elementary
     reflectors of order M
           Q  =  H(k) . . . H(2) H(1)
     as returned by DGEQLF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. M >= N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the (n-k+i)-th column must contain the vector which
              defines the elementary reflector H(i), for i = 1,2,...,k, as
              returned by DGEQLF in the last k columns of its array
              argument A.
              On exit, the M-by-N matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQLF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N).
              For optimum performance LWORK >= N*NB, where NB is the
              optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 127 of file dorgql.f.

  subroutine dorgqr (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DORGQR

Purpose:

     DORGQR generates an M-by-N real matrix Q with orthonormal columns,
     which is defined as the first N columns of a product of K elementary
     reflectors of order M
           Q  =  H(1) H(2) . . . H(k)
     as returned by DGEQRF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. M >= N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the i-th column must contain the vector which
              defines the elementary reflector H(i), for i = 1,2,...,k, as
              returned by DGEQRF in the first k columns of its array
              argument A.
              On exit, the M-by-N matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQRF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N).
              For optimum performance LWORK >= N*NB, where NB is the
              optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 127 of file dorgqr.f.

  subroutine dorgr2 (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)

DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).

Purpose:

     DORGR2 generates an m by n real matrix Q with orthonormal rows,
     which is defined as the last m rows of a product of k elementary
     reflectors of order n
           Q  =  H(1) H(2) . . . H(k)
     as returned by DGERQF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. N >= M.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. M >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the (m-k+i)-th row must contain the vector which
              defines the elementary reflector H(i), for i = 1,2,...,k, as
              returned by DGERQF in the last k rows of its array argument
              A.
              On exit, the m by n matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGERQF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (M)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 113 of file dorgr2.f.

  subroutine dorgrq (integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DORGRQ

Purpose:

     DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
     which is defined as the last M rows of a product of K elementary
     reflectors of order N
           Q  =  H(1) H(2) . . . H(k)
     as returned by DGERQF.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix Q. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q. N >= M.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines the
              matrix Q. M >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the (m-k+i)-th row must contain the vector which
              defines the elementary reflector H(i), for i = 1,2,...,k, as
              returned by DGERQF in the last k rows of its array argument
              A.
              On exit, the M-by-N matrix Q.

    LDA

              LDA is INTEGER
              The first dimension of the array A. LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGERQF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,M).
              For optimum performance LWORK >= M*NB, where NB is the
              optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument has an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 127 of file dorgrq.f.

  subroutine dorgtr (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DORGTR

Purpose:

     DORGTR generates a real orthogonal matrix Q which is defined as the
     product of n-1 elementary reflectors of order N, as returned by
     DSYTRD:
     if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
     if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U': Upper triangle of A contains elementary reflectors
                     from DSYTRD;
              = 'L': Lower triangle of A contains elementary reflectors
                     from DSYTRD.

    N

              N is INTEGER
              The order of the matrix Q. N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the vectors which define the elementary reflectors,
              as returned by DSYTRD.
              On exit, the N-by-N orthogonal matrix Q.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,N).

    TAU

              TAU is DOUBLE PRECISION array, dimension (N-1)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DSYTRD.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N-1).
              For optimum performance LWORK >= (N-1)*NB, where NB is
              the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 122 of file dorgtr.f.

  subroutine dorgtsqr (integer m, integer n, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer lwork, integer info)

DORGTSQR

Purpose:

     DORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns,
     which are the first N columns of a product of real orthogonal
     matrices of order M which are returned by DLATSQR
          Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
     See the documentation for DLATSQR.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. M >= N >= 0.

    MB

              MB is INTEGER
              The row block size used by DLATSQR to return
              arrays A and T. MB > N.
              (Note that if MB > M, then M is used instead of MB
              as the row block size).

    NB

              NB is INTEGER
              The column block size used by DLATSQR to return
              arrays A and T. NB >= 1.
              (Note that if NB > N, then N is used instead of NB
              as the column block size).

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry:
                 The elements on and above the diagonal are not accessed.
                 The elements below the diagonal represent the unit
                 lower-trapezoidal blocked matrix V computed by DLATSQR
                 that defines the input matrices Q_in(k) (ones on the
                 diagonal are not stored) (same format as the output A
                 below the diagonal in DLATSQR).
              On exit:
                 The array A contains an M-by-N orthonormal matrix Q_out,
                 i.e the columns of A are orthogonal unit vectors.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    T

              T is DOUBLE PRECISION array,
              dimension (LDT, N * NIRB)
              where NIRB = Number_of_input_row_blocks
                         = MAX( 1, CEIL((M-N)/(MB-N)) )
              Let NICB = Number_of_input_col_blocks
                       = CEIL(N/NB)
              The upper-triangular block reflectors used to define the
              input matrices Q_in(k), k=(1:NIRB*NICB). The block
              reflectors are stored in compact form in NIRB block
              reflector sequences. Each of NIRB block reflector sequences
              is stored in a larger NB-by-N column block of T and consists
              of NICB smaller NB-by-NB upper-triangular column blocks.
              (same format as the output T in DLATSQR).

    LDT

              LDT is INTEGER
              The leading dimension of the array T.
              LDT >= max(1,min(NB1,N)).

    WORK

              (workspace) DOUBLE PRECISION array, dimension (MAX(2,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              The dimension of the array WORK.  LWORK >= (M+NB)*N.
              If LWORK = -1, then a workspace query is assumed.
              The routine only calculates the optimal size of the WORK
              array, returns this value as the first entry of the WORK
              array, and no error message related to LWORK is issued
              by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

     November 2019, Igor Kozachenko,
                    Computer Science Division,
                    University of California, Berkeley

Definition at line 173 of file dorgtsqr.f.

  subroutine dorgtsqr_row (integer m, integer n, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer lwork, integer info)

DORGTSQR_ROW

Purpose:

     DORGTSQR_ROW generates an M-by-N real matrix Q_out with
     orthonormal columns from the output of DLATSQR. These N orthonormal
     columns are the first N columns of a product of complex unitary
     matrices Q(k)_in of order M, which are returned by DLATSQR in
     a special format.
          Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
     The input matrices Q(k)_in are stored in row and column blocks in A.
     See the documentation of DLATSQR for more details on the format of
     Q(k)_in, where each Q(k)_in is represented by block Householder
     transformations. This routine calls an auxiliary routine DLARFB_GETT,
     where the computation is performed on each individual block. The
     algorithm first sweeps NB-sized column blocks from the right to left
     starting in the bottom row block and continues to the top row block
     (hence _ROW in the routine name). This sweep is in reverse order of
     the order in which DLATSQR generates the output blocks.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. M >= N >= 0.

    MB

              MB is INTEGER
              The row block size used by DLATSQR to return
              arrays A and T. MB > N.
              (Note that if MB > M, then M is used instead of MB
              as the row block size).

    NB

              NB is INTEGER
              The column block size used by DLATSQR to return
              arrays A and T. NB >= 1.
              (Note that if NB > N, then N is used instead of NB
              as the column block size).

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry:
                 The elements on and above the diagonal are not used as
                 input. The elements below the diagonal represent the unit
                 lower-trapezoidal blocked matrix V computed by DLATSQR
                 that defines the input matrices Q_in(k) (ones on the
                 diagonal are not stored). See DLATSQR for more details.
              On exit:
                 The array A contains an M-by-N orthonormal matrix Q_out,
                 i.e the columns of A are orthogonal unit vectors.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    T

              T is DOUBLE PRECISION array,
              dimension (LDT, N * NIRB)
              where NIRB = Number_of_input_row_blocks
                         = MAX( 1, CEIL((M-N)/(MB-N)) )
              Let NICB = Number_of_input_col_blocks
                       = CEIL(N/NB)
              The upper-triangular block reflectors used to define the
              input matrices Q_in(k), k=(1:NIRB*NICB). The block
              reflectors are stored in compact form in NIRB block
              reflector sequences. Each of the NIRB block reflector
              sequences is stored in a larger NB-by-N column block of T
              and consists of NICB smaller NB-by-NB upper-triangular
              column blocks. See DLATSQR for more details on the format
              of T.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.
              LDT >= max(1,min(NB,N)).

    WORK

              (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              The dimension of the array WORK.
              LWORK >= NBLOCAL * MAX(NBLOCAL,(N-NBLOCAL)),
              where NBLOCAL=MIN(NB,N).
              If LWORK = -1, then a workspace query is assumed.
              The routine only calculates the optimal size of the WORK
              array, returns this value as the first entry of the WORK
              array, and no error message related to LWORK is issued
              by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

     November 2020, Igor Kozachenko,
                    Computer Science Division,
                    University of California, Berkeley

Definition at line 186 of file dorgtsqr_row.f.

  subroutine dorhr_col (integer m, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) d, integer info)

DORHR_COL

Purpose:

      DORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
      as input, stored in A, and performs Householder Reconstruction (HR),
      i.e. reconstructs Householder vectors V(i) implicitly representing
      another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
      where S is an N-by-N diagonal matrix with diagonal entries
      equal to +1 or -1. The Householder vectors (columns V(i) of V) are
      stored in A on output, and the diagonal entries of S are stored in D.
      Block reflectors are also returned in T
      (same output format as DGEQRT).

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A. M >= N >= 0.

    NB

              NB is INTEGER
              The column block size to be used in the reconstruction
              of Householder column vector blocks in the array A and
              corresponding block reflectors in the array T. NB >= 1.
              (Note that if NB > N, then N is used instead of NB
              as the column block size.)

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry:
                 The array A contains an M-by-N orthonormal matrix Q_in,
                 i.e the columns of A are orthogonal unit vectors.
              On exit:
                 The elements below the diagonal of A represent the unit
                 lower-trapezoidal matrix V of Householder column vectors
                 V(i). The unit diagonal entries of V are not stored
                 (same format as the output below the diagonal in A from
                 DGEQRT). The matrix T and the matrix V stored on output
                 in A implicitly define Q_out.
                 The elements above the diagonal contain the factor U
                 of the 'modified' LU-decomposition:
                    Q_in - ( S ) = V * U
                           ( 0 )
                 where 0 is a (M-N)-by-(M-N) zero matrix.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    T

              T is DOUBLE PRECISION array,
              dimension (LDT, N)
              Let NOCB = Number_of_output_col_blocks
                       = CEIL(N/NB)
              On exit, T(1:NB, 1:N) contains NOCB upper-triangular
              block reflectors used to define Q_out stored in compact
              form as a sequence of upper-triangular NB-by-NB column
              blocks (same format as the output T in DGEQRT).
              The matrix T and the matrix V stored on output in A
              implicitly define Q_out. NOTE: The lower triangles
              below the upper-triangular blocks will be filled with
              zeros. See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.
              LDT >= max(1,min(NB,N)).

    D

              D is DOUBLE PRECISION array, dimension min(M,N).
              The elements can be only plus or minus one.
              D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
              1 <= i <= min(M,N), and Q_in_i is Q_in after performing
              i-1 steps of “modified” Gaussian elimination.
              See Further Details.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Further Details:

     The computed M-by-M orthogonal factor Q_out is defined implicitly as
     a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
     the compact WY-representation format in the corresponding blocks of
     matrices V (stored in A) and T.
     The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
     matrix A contains the column vectors V(i) in NB-size column
     blocks VB(j). For example, VB(1) contains the columns
     V(1), V(2), ... V(NB). NOTE: The unit entries on
     the diagonal of Y are not stored in A.
     The number of column blocks is
         NOCB = Number_of_output_col_blocks = CEIL(N/NB)
     where each block is of order NB except for the last block, which
     is of order LAST_NB = N - (NOCB-1)*NB.
     For example, if M=6,  N=5 and NB=2, the matrix V is
         V = (    VB(1),   VB(2), VB(3) ) =
           = (   1                      )
             ( v21    1                 )
             ( v31  v32    1            )
             ( v41  v42  v43   1        )
             ( v51  v52  v53  v54    1  )
             ( v61  v62  v63  v54   v65 )
     For each of the column blocks VB(i), an upper-triangular block
     reflector TB(i) is computed. These blocks are stored as
     a sequence of upper-triangular column blocks in the NB-by-N
     matrix T. The size of each TB(i) block is NB-by-NB, except
     for the last block, whose size is LAST_NB-by-LAST_NB.
     For example, if M=6,  N=5 and NB=2, the matrix T is
         T  = (    TB(1),    TB(2), TB(3) ) =
            = ( t11  t12  t13  t14   t15  )
              (      t22       t24        )
     The M-by-M factor Q_out is given as a product of NOCB
     orthogonal M-by-M matrices Q_out(i).
         Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
     where each matrix Q_out(i) is given by the WY-representation
     using corresponding blocks from the matrices V and T:
         Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
     where I is the identity matrix. Here is the formula with matrix
     dimensions:
      Q(i){M-by-M} = I{M-by-M} -
        VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
     where INB = NB, except for the last block NOCB
     for which INB=LAST_NB.
     =====
     NOTE:
     =====
     If Q_in is the result of doing a QR factorization
     B = Q_in * R_in, then:
     B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
     So if one wants to interpret Q_out as the result
     of the QR factorization of B, then the corresponding R_out
     should be equal to R_out = S * R_in, i.e. some rows of R_in
     should be multiplied by -1.
     For the details of the algorithm, see [1].
     [1] 'Reconstructing Householder vectors from tall-skinny QR',
         G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
         E. Solomonik, J. Parallel Distrib. Comput.,
         vol. 85, pp. 3-31, 2015.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

     November   2019, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

Definition at line 258 of file dorhr_col.f.

  subroutine dorm2l (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).

Purpose:

     DORM2L overwrites the general real m by n matrix C with
           Q * C  if SIDE = 'L' and TRANS = 'N', or
           Q**T * C  if SIDE = 'L' and TRANS = 'T', or
           C * Q  if SIDE = 'R' and TRANS = 'N', or
           C * Q**T if SIDE = 'R' and TRANS = 'T',
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(k) . . . H(2) H(1)
     as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left
              = 'R': apply Q or Q**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply Q  (No transpose)
              = 'T': apply Q**T (Transpose)

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,K)
              The i-th column must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGEQLF in the last k columns of its array argument A.
              A is modified by the routine but restored on exit.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,M);
              if SIDE = 'R', LDA >= max(1,N).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQLF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                                       (N) if SIDE = 'L',
                                       (M) if SIDE = 'R'

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 157 of file dorm2l.f.

  subroutine dorm2r (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).

Purpose:

     DORM2R overwrites the general real m by n matrix C with
           Q * C  if SIDE = 'L' and TRANS = 'N', or
           Q**T* C  if SIDE = 'L' and TRANS = 'T', or
           C * Q  if SIDE = 'R' and TRANS = 'N', or
           C * Q**T if SIDE = 'R' and TRANS = 'T',
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(1) H(2) . . . H(k)
     as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left
              = 'R': apply Q or Q**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply Q  (No transpose)
              = 'T': apply Q**T (Transpose)

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,K)
              The i-th column must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGEQRF in the first k columns of its array argument A.
              A is modified by the routine but restored on exit.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,M);
              if SIDE = 'R', LDA >= max(1,N).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQRF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                                       (N) if SIDE = 'L',
                                       (M) if SIDE = 'R'

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 157 of file dorm2r.f.

  subroutine dormbr (character vect, character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMBR

Purpose:

     If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C
     with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
     with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      P * C          C * P
     TRANS = 'T':      P**T * C       C * P**T
     Here Q and P**T are the orthogonal matrices determined by DGEBRD when
     reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
     P**T are defined as products of elementary reflectors H(i) and G(i)
     respectively.
     Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
     order of the orthogonal matrix Q or P**T that is applied.
     If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
     if nq >= k, Q = H(1) H(2) . . . H(k);
     if nq < k, Q = H(1) H(2) . . . H(nq-1).
     If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
     if k < nq, P = G(1) G(2) . . . G(k);
     if k >= nq, P = G(1) G(2) . . . G(nq-1).

Parameters
    VECT

              VECT is CHARACTER*1
              = 'Q': apply Q or Q**T;
              = 'P': apply P or P**T.

    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q, Q**T, P or P**T from the Left;
              = 'R': apply Q, Q**T, P or P**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q  or P;
              = 'T':  Transpose, apply Q**T or P**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              If VECT = 'Q', the number of columns in the original
              matrix reduced by DGEBRD.
              If VECT = 'P', the number of rows in the original
              matrix reduced by DGEBRD.
              K >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                    (LDA,min(nq,K)) if VECT = 'Q'
                                    (LDA,nq)        if VECT = 'P'
              The vectors which define the elementary reflectors H(i) and
              G(i), whose products determine the matrices Q and P, as
              returned by DGEBRD.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If VECT = 'Q', LDA >= max(1,nq);
              if VECT = 'P', LDA >= max(1,min(nq,K)).

    TAU

              TAU is DOUBLE PRECISION array, dimension (min(nq,K))
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i) or G(i) which determines Q or P, as returned
              by DGEBRD in the array argument TAUQ or TAUP.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
              or P*C or P**T*C or C*P or C*P**T.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For optimum performance LWORK >= N*NB if SIDE = 'L', and
              LWORK >= M*NB if SIDE = 'R', where NB is the optimal
              blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 193 of file dormbr.f.

  subroutine dormhr (character side, character trans, integer m, integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMHR

Purpose:

     DORMHR overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix of order nq, with nq = m if
     SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
     IHI-ILO elementary reflectors, as returned by DGEHRD:
     Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI must have the same values as in the previous call
              of DGEHRD. Q is equal to the unit matrix except in the
              submatrix Q(ilo+1:ihi,ilo+1:ihi).
              If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
              ILO = 1 and IHI = 0, if M = 0;
              if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
              ILO = 1 and IHI = 0, if N = 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L'
                                   (LDA,N) if SIDE = 'R'
              The vectors which define the elementary reflectors, as
              returned by DGEHRD.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

    TAU

              TAU is DOUBLE PRECISION array, dimension
                                   (M-1) if SIDE = 'L'
                                   (N-1) if SIDE = 'R'
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEHRD.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For optimum performance LWORK >= N*NB if SIDE = 'L', and
              LWORK >= M*NB if SIDE = 'R', where NB is the optimal
              blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 176 of file dormhr.f.

  subroutine dorml2 (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).

Purpose:

     DORML2 overwrites the general real m by n matrix C with
           Q * C  if SIDE = 'L' and TRANS = 'N', or
           Q**T* C  if SIDE = 'L' and TRANS = 'T', or
           C * Q  if SIDE = 'R' and TRANS = 'N', or
           C * Q**T if SIDE = 'R' and TRANS = 'T',
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(k) . . . H(2) H(1)
     as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left
              = 'R': apply Q or Q**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply Q  (No transpose)
              = 'T': apply Q**T (Transpose)

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L',
                                   (LDA,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGELQF in the first k rows of its array argument A.
              A is modified by the routine but restored on exit.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGELQF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                                       (N) if SIDE = 'L',
                                       (M) if SIDE = 'R'

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 157 of file dorml2.f.

  subroutine dormlq (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMLQ

Purpose:

     DORMLQ overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(k) . . . H(2) H(1)
     as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L',
                                   (LDA,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGELQF in the first k rows of its array argument A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGELQF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 165 of file dormlq.f.

  subroutine dormql (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMQL

Purpose:

     DORMQL overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(k) . . . H(2) H(1)
     as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,K)
              The i-th column must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGEQLF in the last k columns of its array argument A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,M);
              if SIDE = 'R', LDA >= max(1,N).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQLF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 165 of file dormql.f.

  subroutine dormqr (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMQR

Purpose:

     DORMQR overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(1) H(2) . . . H(k)
     as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,K)
              The i-th column must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGEQRF in the first k columns of its array argument A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,M);
              if SIDE = 'R', LDA >= max(1,N).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGEQRF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 165 of file dormqr.f.

  subroutine dormr2 (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).

Purpose:

     DORMR2 overwrites the general real m by n matrix C with
           Q * C  if SIDE = 'L' and TRANS = 'N', or
           Q**T* C  if SIDE = 'L' and TRANS = 'T', or
           C * Q  if SIDE = 'R' and TRANS = 'N', or
           C * Q**T if SIDE = 'R' and TRANS = 'T',
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(1) H(2) . . . H(k)
     as returned by DGERQF. Q is of order m if SIDE = 'L' and of order n
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left
              = 'R': apply Q or Q**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply Q  (No transpose)
              = 'T': apply Q' (Transpose)

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L',
                                   (LDA,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGERQF in the last k rows of its array argument A.
              A is modified by the routine but restored on exit.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGERQF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the m by n matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                                       (N) if SIDE = 'L',
                                       (M) if SIDE = 'R'

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 157 of file dormr2.f.

  subroutine dormr3 (character side, character trans, integer m, integer n, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).

Purpose:

     DORMR3 overwrites the general real m by n matrix C with
           Q * C  if SIDE = 'L' and TRANS = 'N', or
           Q**T* C  if SIDE = 'L' and TRANS = 'C', or
           C * Q  if SIDE = 'R' and TRANS = 'N', or
           C * Q**T if SIDE = 'R' and TRANS = 'C',
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(1) H(2) . . . H(k)
     as returned by DTZRZF. Q is of order m if SIDE = 'L' and of order n
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left
              = 'R': apply Q or Q**T from the Right

    TRANS

              TRANS is CHARACTER*1
              = 'N': apply Q  (No transpose)
              = 'T': apply Q**T (Transpose)

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    L

              L is INTEGER
              The number of columns of the matrix A containing
              the meaningful part of the Householder reflectors.
              If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L',
                                   (LDA,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DTZRZF in the last k rows of its array argument A.
              A is modified by the routine but restored on exit.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DTZRZF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the m-by-n matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension
                                       (N) if SIDE = 'L',
                                       (M) if SIDE = 'R'

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

Definition at line 176 of file dormr3.f.

  subroutine dormrq (character side, character trans, integer m, integer n, integer k, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMRQ

Purpose:

     DORMRQ overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(1) H(2) . . . H(k)
     as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L',
                                   (LDA,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGERQF in the last k rows of its array argument A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DGERQF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 165 of file dormrq.f.

  subroutine dormrz (character side, character trans, integer m, integer n, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMRZ

Purpose:

     DORMRZ overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix defined as the product of k
     elementary reflectors
           Q = H(1) H(2) . . . H(k)
     as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N
     if SIDE = 'R'.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.
              If SIDE = 'L', M >= K >= 0;
              if SIDE = 'R', N >= K >= 0.

    L

              L is INTEGER
              The number of columns of the matrix A containing
              the meaningful part of the Householder reflectors.
              If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L',
                                   (LDA,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DTZRZF in the last k rows of its array argument A.
              A is modified by the routine but restored on exit.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DTZRZF.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

Definition at line 185 of file dormrz.f.

  subroutine dormtr (character side, character uplo, character trans, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DORMTR

Purpose:

     DORMTR overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q * C          C * Q
     TRANS = 'T':      Q**T * C       C * Q**T
     where Q is a real orthogonal matrix of order nq, with nq = m if
     SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
     nq-1 elementary reflectors, as returned by DSYTRD:
     if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
     if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    UPLO

              UPLO is CHARACTER*1
              = 'U': Upper triangle of A contains elementary reflectors
                     from DSYTRD;
              = 'L': Lower triangle of A contains elementary reflectors
                     from DSYTRD.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix C. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix C. N >= 0.

    A

              A is DOUBLE PRECISION array, dimension
                                   (LDA,M) if SIDE = 'L'
                                   (LDA,N) if SIDE = 'R'
              The vectors which define the elementary reflectors, as
              returned by DSYTRD.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

    TAU

              TAU is DOUBLE PRECISION array, dimension
                                   (M-1) if SIDE = 'L'
                                   (N-1) if SIDE = 'R'
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by DSYTRD.

    C

              C is DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

    LDC

              LDC is INTEGER
              The leading dimension of the array C. LDC >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If SIDE = 'L', LWORK >= max(1,N);
              if SIDE = 'R', LWORK >= max(1,M).
              For optimum performance LWORK >= N*NB if SIDE = 'L', and
              LWORK >= M*NB if SIDE = 'R', where NB is the optimal
              blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 169 of file dormtr.f.

  subroutine dpbcon (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision anorm, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DPBCON

Purpose:

     DPBCON estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric positive definite band matrix using the
     Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangular factor stored in AB;
              = 'L':  Lower triangular factor stored in AB.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The triangular factor U or L from the Cholesky factorization
              A = U**T*U or A = L*L**T of the band matrix A, stored in the
              first KD+1 rows of the array.  The j-th column of U or L is
              stored in the j-th column of the array AB as follows:
              if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
              if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    ANORM

              ANORM is DOUBLE PRECISION
              The 1-norm (or infinity-norm) of the symmetric band matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 130 of file dpbcon.f.

  subroutine dpbequ (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) s, double precision scond, double precision amax, integer info)

DPBEQU

Purpose:

     DPBEQU computes row and column scalings intended to equilibrate a
     symmetric positive definite band matrix A and reduce its condition
     number (with respect to the two-norm).  S contains the scale factors,
     S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
     elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
     choice of S puts the condition number of B within a factor N of the
     smallest possible condition number over all possible diagonal
     scalings.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangular of A is stored;
              = 'L':  Lower triangular of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The upper or lower triangle of the symmetric band matrix A,
              stored in the first KD+1 rows of the array.  The j-th column
              of A is stored in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array A.  LDAB >= KD+1.

    S

              S is DOUBLE PRECISION array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

    SCOND

              SCOND is DOUBLE PRECISION
              If INFO = 0, S contains the ratio of the smallest S(i) to
              the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
              large nor too small, it is not worth scaling by S.

    AMAX

              AMAX is DOUBLE PRECISION
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, the i-th diagonal element is nonpositive.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 128 of file dpbequ.f.

  subroutine dpbrfs (character uplo, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldafb, * ) afb, integer ldafb, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DPBRFS

Purpose:

     DPBRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is symmetric positive definite
     and banded, and provides error bounds and backward error estimates
     for the solution.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The upper or lower triangle of the symmetric band matrix A,
              stored in the first KD+1 rows of the array.  The j-th column
              of A is stored in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    AFB

              AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
              The triangular factor U or L from the Cholesky factorization
              A = U**T*U or A = L*L**T of the band matrix A as computed by
              DPBTRF, in the same storage format as A (see AB).

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= KD+1.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by DPBTRS.
              On exit, the improved solution matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 187 of file dpbrfs.f.

  subroutine dpbstf (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, integer info)

DPBSTF

Purpose:

     DPBSTF computes a split Cholesky factorization of a real
     symmetric positive definite band matrix A.
     This routine is designed to be used in conjunction with DSBGST.
     The factorization has the form  A = S**T*S  where S is a band matrix
     of the same bandwidth as A and the following structure:
       S = ( U    )
           ( M  L )
     where U is upper triangular of order m = (n+kd)/2, and L is lower
     triangular of order n-m.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first kd+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, if INFO = 0, the factor S from the split Cholesky
              factorization A = S**T*S. See Further Details.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, the factorization could not be completed,
                   because the updated element a(i,i) was negative; the
                   matrix A is not positive definite.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The band storage scheme is illustrated by the following example, when
      N = 7, KD = 2:
      S = ( s11  s12  s13                     )
          (      s22  s23  s24                )
          (           s33  s34                )
          (                s44                )
          (           s53  s54  s55           )
          (                s64  s65  s66      )
          (                     s75  s76  s77 )
      If UPLO = 'U', the array AB holds:
      on entry:                          on exit:
       *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
       *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
      a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
      If UPLO = 'L', the array AB holds:
      on entry:                          on exit:
      a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
      a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
      a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *
      Array elements marked * are not used by the routine.

Definition at line 151 of file dpbstf.f.

  subroutine dpbtf2 (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, integer info)

DPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

Purpose:

     DPBTF2 computes the Cholesky factorization of a real symmetric
     positive definite band matrix A.
     The factorization has the form
        A = U**T * U ,  if UPLO = 'U', or
        A = L  * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix, U**T is the transpose of U, and
     L is lower triangular.
     This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored:
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of super-diagonals of the matrix A if UPLO = 'U',
              or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, if INFO = 0, the triangular factor U or L from the
              Cholesky factorization A = U**T*U or A = L*L**T of the band
              matrix A, in the same storage format as A.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value
              > 0: if INFO = k, the leading minor of order k is not
                   positive definite, and the factorization could not be
                   completed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The band storage scheme is illustrated by the following example, when
      N = 6, KD = 2, and UPLO = 'U':
      On entry:                       On exit:
          *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
          *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
         a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
      Similarly, if UPLO = 'L' the format of A is as follows:
      On entry:                       On exit:
         a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
         a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
         a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
      Array elements marked * are not used by the routine.

Definition at line 141 of file dpbtf2.f.

  subroutine dpbtrf (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, integer info)

DPBTRF

Purpose:

     DPBTRF computes the Cholesky factorization of a real symmetric
     positive definite band matrix A.
     The factorization has the form
        A = U**T * U,  if UPLO = 'U', or
        A = L  * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is lower triangular.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, if INFO = 0, the triangular factor U or L from the
              Cholesky factorization A = U**T*U or A = L*L**T of the band
              matrix A, in the same storage format as A.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the leading minor of order i is not
                    positive definite, and the factorization could not be
                    completed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The band storage scheme is illustrated by the following example, when
      N = 6, KD = 2, and UPLO = 'U':
      On entry:                       On exit:
          *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
          *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
         a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
      Similarly, if UPLO = 'L' the format of A is as follows:
      On entry:                       On exit:
         a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
         a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
         a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
      Array elements marked * are not used by the routine.

Contributors:
    Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line 141 of file dpbtrf.f.

  subroutine dpbtrs (character uplo, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DPBTRS

Purpose:

     DPBTRS solves a system of linear equations A*X = B with a symmetric
     positive definite band matrix A using the Cholesky factorization
     A = U**T*U or A = L*L**T computed by DPBTRF.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangular factor stored in AB;
              = 'L':  Lower triangular factor stored in AB.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The triangular factor U or L from the Cholesky factorization
              A = U**T*U or A = L*L**T of the band matrix A, stored in the
              first KD+1 rows of the array.  The j-th column of U or L is
              stored in the j-th column of the array AB as follows:
              if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
              if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 120 of file dpbtrs.f.

  subroutine dpftrf (character transr, character uplo, integer n, double precision, dimension( 0: * ) a, integer info)

DPFTRF

Purpose:

     DPFTRF computes the Cholesky factorization of a real symmetric
     positive definite matrix A.
     The factorization has the form
        A = U**T * U,  if UPLO = 'U', or
        A = L  * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is lower triangular.
     This is the block version of the algorithm, calling Level 3 BLAS.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal TRANSR of RFP A is stored;
              = 'T':  The Transpose TRANSR of RFP A is stored.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of RFP A is stored;
              = 'L':  Lower triangle of RFP A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
              On entry, the symmetric matrix A in RFP format. RFP format is
              described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
              then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
              (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
              the transpose of RFP A as defined when
              TRANSR = 'N'. The contents of RFP A are defined by UPLO as
              follows: If UPLO = 'U' the RFP A contains the NT elements of
              upper packed A. If UPLO = 'L' the RFP A contains the elements
              of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
              'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
              is odd. See the Note below for more details.
              On exit, if INFO = 0, the factor U or L from the Cholesky
              factorization RFP A = U**T*U or RFP A = L*L**T.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the leading minor of order i is not
                    positive definite, and the factorization could not be
                    completed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 197 of file dpftrf.f.

  subroutine dpftri (character transr, character uplo, integer n, double precision, dimension( 0: * ) a, integer info)

DPFTRI

Purpose:

     DPFTRI computes the inverse of a (real) symmetric positive definite
     matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
     computed by DPFTRF.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal TRANSR of RFP A is stored;
              = 'T':  The Transpose TRANSR of RFP A is stored.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
              On entry, the symmetric matrix A in RFP format. RFP format is
              described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
              then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
              (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
              the transpose of RFP A as defined when
              TRANSR = 'N'. The contents of RFP A are defined by UPLO as
              follows: If UPLO = 'U' the RFP A contains the nt elements of
              upper packed A. If UPLO = 'L' the RFP A contains the elements
              of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
              'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
              is odd. See the Note below for more details.
              On exit, the symmetric inverse of the original matrix, in the
              same storage format.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the (i,i) element of the factor U or L is
                    zero, and the inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 190 of file dpftri.f.

  subroutine dpftrs (character transr, character uplo, integer n, integer nrhs, double precision, dimension( 0: * ) a, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DPFTRS

Purpose:

     DPFTRS solves a system of linear equations A*X = B with a symmetric
     positive definite matrix A using the Cholesky factorization
     A = U**T*U or A = L*L**T computed by DPFTRF.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal TRANSR of RFP A is stored;
              = 'T':  The Transpose TRANSR of RFP A is stored.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of RFP A is stored;
              = 'L':  Lower triangle of RFP A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ).
              The triangular factor U or L from the Cholesky factorization
              of RFP A = U**T*U or RFP A = L*L**T, as computed by DPFTRF.
              See note below for more details about RFP A.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 198 of file dpftrs.f.

  subroutine dppcon (character uplo, integer n, double precision, dimension( * ) ap, double precision anorm, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DPPCON

Purpose:

     DPPCON estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric positive definite packed matrix using
     the Cholesky factorization A = U**T*U or A = L*L**T computed by
     DPPTRF.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The triangular factor U or L from the Cholesky factorization
              A = U**T*U or A = L*L**T, packed columnwise in a linear
              array.  The j-th column of U or L is stored in the array AP
              as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

    ANORM

              ANORM is DOUBLE PRECISION
              The 1-norm (or infinity-norm) of the symmetric matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 117 of file dppcon.f.

  subroutine dppequ (character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) s, double precision scond, double precision amax, integer info)

DPPEQU

Purpose:

     DPPEQU computes row and column scalings intended to equilibrate a
     symmetric positive definite matrix A in packed storage and reduce
     its condition number (with respect to the two-norm).  S contains the
     scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
     B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
     This choice of S puts the condition number of B within a factor N of
     the smallest possible condition number over all possible diagonal
     scalings.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangle of the symmetric matrix A, packed
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    S

              S is DOUBLE PRECISION array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

    SCOND

              SCOND is DOUBLE PRECISION
              If INFO = 0, S contains the ratio of the smallest S(i) to
              the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
              large nor too small, it is not worth scaling by S.

    AMAX

              AMAX is DOUBLE PRECISION
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the i-th diagonal element is nonpositive.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 115 of file dppequ.f.

  subroutine dpprfs (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DPPRFS

Purpose:

     DPPRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is symmetric positive definite
     and packed, and provides error bounds and backward error estimates
     for the solution.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangle of the symmetric matrix A, packed
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    AFP

              AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The triangular factor U or L from the Cholesky factorization
              A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
              packed columnwise in a linear array in the same format as A
              (see AP).

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by DPPTRS.
              On exit, the improved solution matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 169 of file dpprfs.f.

  subroutine dpptrf (character uplo, integer n, double precision, dimension( * ) ap, integer info)

DPPTRF

Purpose:

     DPPTRF computes the Cholesky factorization of a real symmetric
     positive definite matrix A stored in packed format.
     The factorization has the form
        A = U**T * U,  if UPLO = 'U', or
        A = L  * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is lower triangular.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.
              On exit, if INFO = 0, the triangular factor U or L from the
              Cholesky factorization A = U**T*U or A = L*L**T, in the same
              storage format as A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the leading minor of order i is not
                    positive definite, and the factorization could not be
                    completed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the symmetric matrix A:
         a11 a12 a13 a14
             a22 a23 a24
                 a33 a34     (aij = aji)
                     a44
      Packed storage of the upper triangle of A:
      AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 118 of file dpptrf.f.

  subroutine dpptri (character uplo, integer n, double precision, dimension( * ) ap, integer info)

DPPTRI

Purpose:

     DPPTRI computes the inverse of a real symmetric positive definite
     matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
     computed by DPPTRF.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangular factor is stored in AP;
              = 'L':  Lower triangular factor is stored in AP.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the triangular factor U or L from the Cholesky
              factorization A = U**T*U or A = L*L**T, packed columnwise as
              a linear array.  The j-th column of U or L is stored in the
              array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
              On exit, the upper or lower triangle of the (symmetric)
              inverse of A, overwriting the input factor U or L.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the (i,i) element of the factor U or L is
                    zero, and the inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 92 of file dpptri.f.

  subroutine dpptrs (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DPPTRS

Purpose:

     DPPTRS solves a system of linear equations A*X = B with a symmetric
     positive definite matrix A in packed storage using the Cholesky
     factorization A = U**T*U or A = L*L**T computed by DPPTRF.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The triangular factor U or L from the Cholesky factorization
              A = U**T*U or A = L*L**T, packed columnwise in a linear
              array.  The j-th column of U or L is stored in the array AP
              as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 107 of file dpptrs.f.

  subroutine dpstf2 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, integer, dimension( n ) piv, integer rank, double precision tol, double precision, dimension( 2*n ) work, integer info)

DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

Purpose:

     DPSTF2 computes the Cholesky factorization with complete
     pivoting of a real symmetric positive semidefinite matrix A.
     The factorization has the form
        P**T * A * P = U**T * U ,  if UPLO = 'U',
        P**T * A * P = L  * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is lower triangular, and
     P is stored as vector PIV.
     This algorithm does not attempt to check that A is positive
     semidefinite. This version of the algorithm calls level 2 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              n by n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if INFO = 0, the factor U or L from the Cholesky
              factorization as above.

    PIV

              PIV is INTEGER array, dimension (N)
              PIV is such that the nonzero entries are P( PIV(K), K ) = 1.

    RANK

              RANK is INTEGER
              The rank of A given by the number of steps the algorithm
              completed.

    TOL

              TOL is DOUBLE PRECISION
              User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
              will be used. The algorithm terminates at the (K-1)st step
              if the pivot <= TOL.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)
              Work space.

    INFO

              INFO is INTEGER
              < 0: If INFO = -K, the K-th argument had an illegal value,
              = 0: algorithm completed successfully, and
              > 0: the matrix A is either rank deficient with computed rank
                   as returned in RANK, or is not positive semidefinite. See
                   Section 7 of LAPACK Working Note #161 for further
                   information.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 140 of file dpstf2.f.

  subroutine dpstrf (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, integer, dimension( n ) piv, integer rank, double precision tol, double precision, dimension( 2*n ) work, integer info)

DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

Purpose:

     DPSTRF computes the Cholesky factorization with complete
     pivoting of a real symmetric positive semidefinite matrix A.
     The factorization has the form
        P**T * A * P = U**T * U ,  if UPLO = 'U',
        P**T * A * P = L  * L**T,  if UPLO = 'L',
     where U is an upper triangular matrix and L is lower triangular, and
     P is stored as vector PIV.
     This algorithm does not attempt to check that A is positive
     semidefinite. This version of the algorithm calls level 3 BLAS.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              symmetric matrix A is stored.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the symmetric matrix A.  If UPLO = 'U', the leading
              n by n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, if INFO = 0, the factor U or L from the Cholesky
              factorization as above.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    PIV

              PIV is INTEGER array, dimension (N)
              PIV is such that the nonzero entries are P( PIV(K), K ) = 1.

    RANK

              RANK is INTEGER
              The rank of A given by the number of steps the algorithm
              completed.

    TOL

              TOL is DOUBLE PRECISION
              User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
              will be used. The algorithm terminates at the (K-1)st step
              if the pivot <= TOL.

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)
              Work space.

    INFO

              INFO is INTEGER
              < 0: If INFO = -K, the K-th argument had an illegal value,
              = 0: algorithm completed successfully, and
              > 0: the matrix A is either rank deficient with computed rank
                   as returned in RANK, or is not positive semidefinite. See
                   Section 7 of LAPACK Working Note #161 for further
                   information.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 141 of file dpstrf.f.

  subroutine dsbgst (character vect, character uplo, integer n, integer ka, integer kb, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) work, integer info)

DSBGST

Purpose:

     DSBGST reduces a real symmetric-definite banded generalized
     eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
     such that C has the same bandwidth as A.
     B must have been previously factorized as S**T*S by DPBSTF, using a
     split Cholesky factorization. A is overwritten by C = X**T*A*X, where
     X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
     bandwidth of A.

Parameters
    VECT

              VECT is CHARACTER*1
              = 'N':  do not form the transformation matrix X;
              = 'V':  form X.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    KA

              KA is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

    KB

              KB is INTEGER
              The number of superdiagonals of the matrix B if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KA >= KB >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first ka+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
              On exit, the transformed matrix X**T*A*X, stored in the same
              format as A.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KA+1.

    BB

              BB is DOUBLE PRECISION array, dimension (LDBB,N)
              The banded factor S from the split Cholesky factorization of
              B, as returned by DPBSTF, stored in the first KB+1 rows of
              the array.

    LDBB

              LDBB is INTEGER
              The leading dimension of the array BB.  LDBB >= KB+1.

    X

              X is DOUBLE PRECISION array, dimension (LDX,N)
              If VECT = 'V', the n-by-n matrix X.
              If VECT = 'N', the array X is not referenced.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.
              LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 157 of file dsbgst.f.

  subroutine dsbtrd (character vect, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer info)

DSBTRD

Purpose:

     DSBTRD reduces a real symmetric band matrix A to symmetric
     tridiagonal form T by an orthogonal similarity transformation:
     Q**T * A * Q = T.

Parameters
    VECT

              VECT is CHARACTER*1
              = 'N':  do not form Q;
              = 'V':  form Q;
              = 'U':  update a matrix X, by forming X*Q.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, the diagonal elements of AB are overwritten by the
              diagonal elements of the tridiagonal matrix T; if KD > 0, the
              elements on the first superdiagonal (if UPLO = 'U') or the
              first subdiagonal (if UPLO = 'L') are overwritten by the
              off-diagonal elements of T; the rest of AB is overwritten by
              values generated during the reduction.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    D

              D is DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if VECT = 'U', then Q must contain an N-by-N
              matrix X; if VECT = 'N' or 'V', then Q need not be set.
              On exit:
              if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
              if VECT = 'U', Q contains the product X*Q;
              if VECT = 'N', the array Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Modified by Linda Kaufman, Bell Labs.

Definition at line 161 of file dsbtrd.f.

  subroutine dsfrk (character transr, character uplo, character trans, integer n, integer k, double precision alpha, double precision, dimension( lda, * ) a, integer lda, double precision beta, double precision, dimension( * ) c)

DSFRK performs a symmetric rank-k operation for matrix in RFP format.

Purpose:

     Level 3 BLAS like routine for C in RFP Format.
     DSFRK performs one of the symmetric rank--k operations
        C := alpha*A*A**T + beta*C,
     or
        C := alpha*A**T*A + beta*C,
     where alpha and beta are real scalars, C is an n--by--n symmetric
     matrix and A is an n--by--k matrix in the first case and a k--by--n
     matrix in the second case.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal Form of RFP A is stored;
              = 'T':  The Transpose Form of RFP A is stored.

    UPLO

              UPLO is CHARACTER*1
               On  entry, UPLO specifies whether the upper or lower
               triangular part of the array C is to be referenced as
               follows:
                  UPLO = 'U' or 'u'   Only the upper triangular part of C
                                      is to be referenced.
                  UPLO = 'L' or 'l'   Only the lower triangular part of C
                                      is to be referenced.
               Unchanged on exit.

    TRANS

              TRANS is CHARACTER*1
               On entry, TRANS specifies the operation to be performed as
               follows:
                  TRANS = 'N' or 'n'   C := alpha*A*A**T + beta*C.
                  TRANS = 'T' or 't'   C := alpha*A**T*A + beta*C.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the order of the matrix C. N must be
               at least zero.
               Unchanged on exit.

    K

              K is INTEGER
               On entry with TRANS = 'N' or 'n', K specifies the number
               of  columns of the matrix A, and on entry with TRANS = 'T'
               or 't', K specifies the number of rows of the matrix A. K
               must be at least zero.
               Unchanged on exit.

    ALPHA

              ALPHA is DOUBLE PRECISION
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    A

              A is DOUBLE PRECISION array, dimension (LDA,ka)
               where KA
               is K  when TRANS = 'N' or 'n', and is N otherwise. Before
               entry with TRANS = 'N' or 'n', the leading N--by--K part of
               the array A must contain the matrix A, otherwise the leading
               K--by--N part of the array A must contain the matrix A.
               Unchanged on exit.

    LDA

              LDA is INTEGER
               On entry, LDA specifies the first dimension of A as declared
               in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
               then  LDA must be at least  max( 1, n ), otherwise  LDA must
               be at least  max( 1, k ).
               Unchanged on exit.

    BETA

              BETA is DOUBLE PRECISION
               On entry, BETA specifies the scalar beta.
               Unchanged on exit.

    C

              C is DOUBLE PRECISION array, dimension (NT)
               NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
               Format. RFP Format is described by TRANSR, UPLO and N.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 164 of file dsfrk.f.

  subroutine dspcon (character uplo, integer n, double precision, dimension( * ) ap, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSPCON

Purpose:

     DSPCON estimates the reciprocal of the condition number (in the
     1-norm) of a real symmetric packed matrix A using the factorization
     A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are stored
              as an upper or lower triangular matrix.
              = 'U':  Upper triangular, form is A = U*D*U**T;
              = 'L':  Lower triangular, form is A = L*D*L**T.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by DSPTRF, stored as a
              packed triangular matrix.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by DSPTRF.

    ANORM

              ANORM is DOUBLE PRECISION
              The 1-norm of the original matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
              estimate of the 1-norm of inv(A) computed in this routine.

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 123 of file dspcon.f.

  subroutine dspgst (integer itype, character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) bp, integer info)

DSPGST

Purpose:

     DSPGST reduces a real symmetric-definite generalized eigenproblem
     to standard form, using packed storage.
     If ITYPE = 1, the problem is A*x = lambda*B*x,
     and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
     If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
     B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
     B must have been previously factorized as U**T*U or L*L**T by DPPTRF.

Parameters
    ITYPE

              ITYPE is INTEGER
              = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
              = 2 or 3: compute U*A*U**T or L**T*A*L.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored and B is factored as
                      U**T*U;
              = 'L':  Lower triangle of A is stored and B is factored as
                      L*L**T.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              On exit, if INFO = 0, the transformed matrix, stored in the
              same format as A.

    BP

              BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The triangular factor from the Cholesky factorization of B,
              stored in the same format as A, as returned by DPPTRF.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 112 of file dspgst.f.

  subroutine dsprfs (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, integer, dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSPRFS

Purpose:

     DSPRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is symmetric indefinite
     and packed, and provides error bounds and backward error estimates
     for the solution.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangle of the symmetric matrix A, packed
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

    AFP

              AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The factored form of the matrix A.  AFP contains the block
              diagonal matrix D and the multipliers used to obtain the
              factor U or L from the factorization A = U*D*U**T or
              A = L*D*L**T as computed by DSPTRF, stored as a packed
              triangular matrix.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by DSPTRF.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by DSPTRS.
              On exit, the improved solution matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 177 of file dsprfs.f.

  subroutine dsptrd (character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, integer info)

DSPTRD

Purpose:

     DSPTRD reduces a real symmetric matrix A stored in packed form to
     symmetric tridiagonal form T by an orthogonal similarity
     transformation: Q**T * A * Q = T.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, if UPLO = 'U', the diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, and the elements below the first subdiagonal, with
              the array TAU, represent the orthogonal matrix Q as a product
              of elementary reflectors. See Further Details.

    D

              D is DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is DOUBLE PRECISION array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(1).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
      overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-1).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
      overwriting A(i+2:n,i), and tau is stored in TAU(i).

Definition at line 149 of file dsptrd.f.

  subroutine dsptrf (character uplo, integer n, double precision, dimension( * ) ap, integer, dimension( * ) ipiv, integer info)

DSPTRF

Purpose:

     DSPTRF computes the factorization of a real symmetric matrix A stored
     in packed format using the Bunch-Kaufman diagonal pivoting method:
        A = U*D*U**T  or  A = L*D*L**T
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, and D is symmetric and block diagonal with
     1-by-1 and 2-by-2 diagonal blocks.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L, stored as a packed triangular
              matrix overwriting A (see below for further details).

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If IPIV(k) > 0, then rows and columns k and IPIV(k) were
              interchanged and D(k,k) is a 1-by-1 diagonal block.
              If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
              columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
              is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
              IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
              interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
                   has been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if it
                   is used to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', then A = U*D*U**T, where
         U = P(n)*U(n)* ... *P(k)U(k)* ...,
      i.e., U is a product of terms P(k)*U(k), where k decreases from n to
      1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    v    0   )   k-s
         U(k) =  (   0    I    0   )   s
                 (   0    0    I   )   n-k
                    k-s   s   n-k
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
      If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
      and A(k,k), and v overwrites A(1:k-2,k-1:k).
      If UPLO = 'L', then A = L*D*L**T, where
         L = P(1)*L(1)* ... *P(k)*L(k)* ...,
      i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
      n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
      and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
      defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
      that if the diagonal block D(k) is of order s (s = 1 or 2), then
                 (   I    0     0   )  k-1
         L(k) =  (   0    I     0   )  s
                 (   0    v     I   )  n-k-s+1
                    k-1   s  n-k-s+1
      If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
      If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
      and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:
    J. Lewis, Boeing Computer Services Company

Definition at line 158 of file dsptrf.f.

  subroutine dsptri (character uplo, integer n, double precision, dimension( * ) ap, integer, dimension( * ) ipiv, double precision, dimension( * ) work, integer info)

DSPTRI

Purpose:

     DSPTRI computes the inverse of a real symmetric indefinite matrix
     A in packed storage using the factorization A = U*D*U**T or
     A = L*D*L**T computed by DSPTRF.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are stored
              as an upper or lower triangular matrix.
              = 'U':  Upper triangular, form is A = U*D*U**T;
              = 'L':  Lower triangular, form is A = L*D*L**T.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the block diagonal matrix D and the multipliers
              used to obtain the factor U or L as computed by DSPTRF,
              stored as a packed triangular matrix.
              On exit, if INFO = 0, the (symmetric) inverse of the original
              matrix, stored as a packed triangular matrix. The j-th column
              of inv(A) is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
              if UPLO = 'L',
                 AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by DSPTRF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                   inverse could not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 108 of file dsptri.f.

  subroutine dsptrs (character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, integer, dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DSPTRS

Purpose:

     DSPTRS solves a system of linear equations A*X = B with a real
     symmetric matrix A stored in packed format using the factorization
     A = U*D*U**T or A = L*D*L**T computed by DSPTRF.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the details of the factorization are stored
              as an upper or lower triangular matrix.
              = 'U':  Upper triangular, form is A = U*D*U**T;
              = 'L':  Lower triangular, form is A = L*D*L**T.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by DSPTRF, stored as a
              packed triangular matrix.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D
              as determined by DSPTRF.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 114 of file dsptrs.f.

  subroutine dstegr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSTEGR

Purpose:

     DSTEGR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
     a well defined set of pairwise different real eigenvalues, the corresponding
     real eigenvectors are pairwise orthogonal.
     The spectrum may be computed either completely or partially by specifying
     either an interval (VL,VU] or a range of indices IL:IU for the desired
     eigenvalues.
     DSTEGR is a compatibility wrapper around the improved DSTEMR routine.
     See DSTEMR for further details.
     One important change is that the ABSTOL parameter no longer provides any
     benefit and hence is no longer used.
     Note : DSTEGR and DSTEMR work only on machines which follow
     IEEE-754 floating-point standard in their handling of infinities and
     NaNs.  Normal execution may create these exceptiona values and hence
     may abort due to a floating point exception in environments which
     do not conform to the IEEE-754 standard.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

              RANGE is CHARACTER*1
              = 'A': all eigenvalues will be found.
              = 'V': all eigenvalues in the half-open interval (VL,VU]
                     will be found.
              = 'I': the IL-th through IU-th eigenvalues will be found.

    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is DOUBLE PRECISION array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the tridiagonal
              matrix T in elements 1 to N-1 of E. E(N) need not be set on
              input, but is used internally as workspace.
              On exit, E is overwritten.

    VL

              VL is DOUBLE PRECISION
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is DOUBLE PRECISION
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0.
              Not referenced if RANGE = 'A' or 'V'.

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              Unused.  Was the absolute error tolerance for the
              eigenvalues/eigenvectors in previous versions.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is DOUBLE PRECISION array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
              If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
              contain the orthonormal eigenvectors of the matrix T
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.
              Supplying N columns is always safe.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', then LDZ >= max(1,N).

    ISUPPZ

              ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
              The support of the eigenvectors in Z, i.e., the indices
              indicating the nonzero elements in Z. The i-th computed eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ). This is relevant in the case when the matrix
              is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal
              (and minimal) LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,18*N)
              if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (LIWORK)
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= max(1,10*N)
              if the eigenvectors are desired, and LIWORK >= max(1,8*N)
              if only the eigenvalues are to be computed.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              On exit, INFO
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = 1X, internal error in DLARRE,
                    if INFO = 2X, internal error in DLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by DLARRE or
                    DLARRV, respectively.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, LBNL/NERSC, USA

Definition at line 262 of file dstegr.f.

  subroutine dstein (integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, integer m, double precision, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

DSTEIN

Purpose:

     DSTEIN computes the eigenvectors of a real symmetric tridiagonal
     matrix T corresponding to specified eigenvalues, using inverse
     iteration.
     The maximum number of iterations allowed for each eigenvector is
     specified by an internal parameter MAXITS (currently set to 5).

Parameters
    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the tridiagonal matrix T.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The (n-1) subdiagonal elements of the tridiagonal matrix
              T, in elements 1 to N-1.

    M

              M is INTEGER
              The number of eigenvectors to be found.  0 <= M <= N.

    W

              W is DOUBLE PRECISION array, dimension (N)
              The first M elements of W contain the eigenvalues for
              which eigenvectors are to be computed.  The eigenvalues
              should be grouped by split-off block and ordered from
              smallest to largest within the block.  ( The output array
              W from DSTEBZ with ORDER = 'B' is expected here. )

    IBLOCK

              IBLOCK is INTEGER array, dimension (N)
              The submatrix indices associated with the corresponding
              eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
              the first submatrix from the top, =2 if W(i) belongs to
              the second submatrix, etc.  ( The output array IBLOCK
              from DSTEBZ is expected here. )

    ISPLIT

              ISPLIT is INTEGER array, dimension (N)
              The splitting points, at which T breaks up into submatrices.
              The first submatrix consists of rows/columns 1 to
              ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
              through ISPLIT( 2 ), etc.
              ( The output array ISPLIT from DSTEBZ is expected here. )

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, M)
              The computed eigenvectors.  The eigenvector associated
              with the eigenvalue W(i) is stored in the i-th column of
              Z.  Any vector which fails to converge is set to its current
              iterate after MAXITS iterations.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= max(1,N).

    WORK

              WORK is DOUBLE PRECISION array, dimension (5*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    IFAIL

              IFAIL is INTEGER array, dimension (M)
              On normal exit, all elements of IFAIL are zero.
              If one or more eigenvectors fail to converge after
              MAXITS iterations, then their indices are stored in
              array IFAIL.

    INFO

              INFO is INTEGER
              = 0: successful exit.
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, then i eigenvectors failed to converge
                   in MAXITS iterations.  Their indices are stored in
                   array IFAIL.

Internal Parameters:

      MAXITS  INTEGER, default = 5
              The maximum number of iterations performed.
      EXTRA   INTEGER, default = 2
              The number of iterations performed after norm growth
              criterion is satisfied, should be at least 1.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 172 of file dstein.f.

  subroutine dstemr (character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSTEMR

Purpose:

     DSTEMR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
     a well defined set of pairwise different real eigenvalues, the corresponding
     real eigenvectors are pairwise orthogonal.
     The spectrum may be computed either completely or partially by specifying
     either an interval (VL,VU] or a range of indices IL:IU for the desired
     eigenvalues.
     Depending on the number of desired eigenvalues, these are computed either
     by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
     computed by the use of various suitable L D L^T factorizations near clusters
     of close eigenvalues (referred to as RRRs, Relatively Robust
     Representations). An informal sketch of the algorithm follows.
     For each unreduced block (submatrix) of T,
        (a) Compute T - sigma I  = L D L^T, so that L and D
            define all the wanted eigenvalues to high relative accuracy.
            This means that small relative changes in the entries of D and L
            cause only small relative changes in the eigenvalues and
            eigenvectors. The standard (unfactored) representation of the
            tridiagonal matrix T does not have this property in general.
        (b) Compute the eigenvalues to suitable accuracy.
            If the eigenvectors are desired, the algorithm attains full
            accuracy of the computed eigenvalues only right before
            the corresponding vectors have to be computed, see steps c) and d).
        (c) For each cluster of close eigenvalues, select a new
            shift close to the cluster, find a new factorization, and refine
            the shifted eigenvalues to suitable accuracy.
        (d) For each eigenvalue with a large enough relative separation compute
            the corresponding eigenvector by forming a rank revealing twisted
            factorization. Go back to (c) for any clusters that remain.
     For more details, see:
     - Inderjit S. Dhillon and Beresford N. Parlett: 'Multiple representations
       to compute orthogonal eigenvectors of symmetric tridiagonal matrices,'
       Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
     - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and
       Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25,
       2004.  Also LAPACK Working Note 154.
     - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric
       tridiagonal eigenvalue/eigenvector problem',
       Computer Science Division Technical Report No. UCB/CSD-97-971,
       UC Berkeley, May 1997.
     Further Details
     1.DSTEMR works only on machines which follow IEEE-754
     floating-point standard in their handling of infinities and NaNs.
     This permits the use of efficient inner loops avoiding a check for
     zero divisors.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

              RANGE is CHARACTER*1
              = 'A': all eigenvalues will be found.
              = 'V': all eigenvalues in the half-open interval (VL,VU]
                     will be found.
              = 'I': the IL-th through IU-th eigenvalues will be found.

    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is DOUBLE PRECISION array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the tridiagonal
              matrix T in elements 1 to N-1 of E. E(N) need not be set on
              input, but is used internally as workspace.
              On exit, E is overwritten.

    VL

              VL is DOUBLE PRECISION
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is DOUBLE PRECISION
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues. VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0.
              Not referenced if RANGE = 'A' or 'V'.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is DOUBLE PRECISION array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
              If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
              contain the orthonormal eigenvectors of the matrix T
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and can be computed with a workspace
              query by setting NZC = -1, see below.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', then LDZ >= max(1,N).

    NZC

              NZC is INTEGER
              The number of eigenvectors to be held in the array Z.
              If RANGE = 'A', then NZC >= max(1,N).
              If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
              If RANGE = 'I', then NZC >= IU-IL+1.
              If NZC = -1, then a workspace query is assumed; the
              routine calculates the number of columns of the array Z that
              are needed to hold the eigenvectors.
              This value is returned as the first entry of the Z array, and
              no error message related to NZC is issued by XERBLA.

    ISUPPZ

              ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
              The support of the eigenvectors in Z, i.e., the indices
              indicating the nonzero elements in Z. The i-th computed eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ). This is relevant in the case when the matrix
              is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

    TRYRAC

              TRYRAC is LOGICAL
              If TRYRAC = .TRUE., indicates that the code should check whether
              the tridiagonal matrix defines its eigenvalues to high relative
              accuracy.  If so, the code uses relative-accuracy preserving
              algorithms that might be (a bit) slower depending on the matrix.
              If the matrix does not define its eigenvalues to high relative
              accuracy, the code can uses possibly faster algorithms.
              If TRYRAC = .FALSE., the code is not required to guarantee
              relatively accurate eigenvalues and can use the fastest possible
              techniques.
              On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
              does not define its eigenvalues to high relative accuracy.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal
              (and minimal) LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,18*N)
              if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (LIWORK)
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= max(1,10*N)
              if the eigenvectors are desired, and LIWORK >= max(1,8*N)
              if only the eigenvalues are to be computed.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              On exit, INFO
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = 1X, internal error in DLARRE,
                    if INFO = 2X, internal error in DLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by DLARRE or
                    DLARRV, respectively.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Beresford Parlett, University of California, Berkeley, USA
     Jim Demmel, University of California, Berkeley, USA
     Inderjit Dhillon, University of Texas, Austin, USA
     Osni Marques, LBNL/NERSC, USA
     Christof Voemel, University of California, Berkeley, USA

Definition at line 318 of file dstemr.f.

  subroutine dtbcon (character norm, character uplo, character diag, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DTBCON

Purpose:

     DTBCON estimates the reciprocal of the condition number of a
     triangular band matrix A, in either the 1-norm or the infinity-norm.
     The norm of A is computed and an estimate is obtained for
     norm(inv(A)), then the reciprocal of the condition number is
     computed as
        RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies whether the 1-norm condition number or the
              infinity-norm condition number is required:
              = '1' or 'O':  1-norm;
              = 'I':         Infinity-norm.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals or subdiagonals of the
              triangular band matrix A.  KD >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The upper or lower triangular band matrix A, stored in the
              first kd+1 rows of the array. The j-th column of A is stored
              in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              If DIAG = 'U', the diagonal elements of A are not referenced
              and are assumed to be 1.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 141 of file dtbcon.f.

  subroutine dtbrfs (character uplo, character trans, character diag, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DTBRFS

Purpose:

     DTBRFS provides error bounds and backward error estimates for the
     solution to a system of linear equations with a triangular band
     coefficient matrix.
     The solution matrix X must be computed by DTBTRS or some other
     means before entering this routine.  DTBRFS does not do iterative
     refinement because doing so cannot improve the backward error.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B  (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals or subdiagonals of the
              triangular band matrix A.  KD >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The upper or lower triangular band matrix A, stored in the
              first kd+1 rows of the array. The j-th column of A is stored
              in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              If DIAG = 'U', the diagonal elements of A are not referenced
              and are assumed to be 1.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              The solution matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 186 of file dtbrfs.f.

  subroutine dtbtrs (character uplo, character trans, character diag, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DTBTRS

Purpose:

     DTBTRS solves a triangular system of the form
        A * X = B  or  A**T * X = B,
     where A is a triangular band matrix of order N, and B is an
     N-by NRHS matrix.  A check is made to verify that A is nonsingular.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    TRANS

              TRANS is CHARACTER*1
              Specifies the form the system of equations:
              = 'N':  A * X = B  (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    KD

              KD is INTEGER
              The number of superdiagonals or subdiagonals of the
              triangular band matrix A.  KD >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    AB

              AB is DOUBLE PRECISION array, dimension (LDAB,N)
              The upper or lower triangular band matrix A, stored in the
              first kd+1 rows of AB.  The j-th column of A is stored
              in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              If DIAG = 'U', the diagonal elements of A are not referenced
              and are assumed to be 1.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, if INFO = 0, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the i-th diagonal element of A is zero,
                    indicating that the matrix is singular and the
                    solutions X have not been computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 144 of file dtbtrs.f.

  subroutine dtfsm (character transr, character side, character uplo, character trans, character diag, integer m, integer n, double precision alpha, double precision, dimension( 0: * ) a, double precision, dimension( 0: ldb-1, 0: * ) b, integer ldb)

DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

     Level 3 BLAS like routine for A in RFP Format.
     DTFSM  solves the matrix equation
        op( A )*X = alpha*B  or  X*op( A ) = alpha*B
     where alpha is a scalar, X and B are m by n matrices, A is a unit, or
     non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
        op( A ) = A   or   op( A ) = A**T.
     A is in Rectangular Full Packed (RFP) Format.
     The matrix X is overwritten on B.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal Form of RFP A is stored;
              = 'T':  The Transpose Form of RFP A is stored.

    SIDE

              SIDE is CHARACTER*1
               On entry, SIDE specifies whether op( A ) appears on the left
               or right of X as follows:
                  SIDE = 'L' or 'l'   op( A )*X = alpha*B.
                  SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
               Unchanged on exit.

    UPLO

              UPLO is CHARACTER*1
               On entry, UPLO specifies whether the RFP matrix A came from
               an upper or lower triangular matrix as follows:
               UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
               UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix
               Unchanged on exit.

    TRANS

              TRANS is CHARACTER*1
               On entry, TRANS  specifies the form of op( A ) to be used
               in the matrix multiplication as follows:
                  TRANS  = 'N' or 'n'   op( A ) = A.
                  TRANS  = 'T' or 't'   op( A ) = A'.
               Unchanged on exit.

    DIAG

              DIAG is CHARACTER*1
               On entry, DIAG specifies whether or not RFP A is unit
               triangular as follows:
                  DIAG = 'U' or 'u'   A is assumed to be unit triangular.
                  DIAG = 'N' or 'n'   A is not assumed to be unit
                                      triangular.
               Unchanged on exit.

    M

              M is INTEGER
               On entry, M specifies the number of rows of B. M must be at
               least zero.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the number of columns of B.  N must be
               at least zero.
               Unchanged on exit.

    ALPHA

              ALPHA is DOUBLE PRECISION
               On entry,  ALPHA specifies the scalar  alpha. When  alpha is
               zero then  A is not referenced and  B need not be set before
               entry.
               Unchanged on exit.

    A

              A is DOUBLE PRECISION array, dimension (NT)
               NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
               RFP Format is described by TRANSR, UPLO and N as follows:
               If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
               K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
               TRANSR = 'T' then RFP is the transpose of RFP A as
               defined when TRANSR = 'N'. The contents of RFP A are defined
               by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
               elements of upper packed A either in normal or
               transpose Format. If UPLO = 'L' the RFP A contains
               the NT elements of lower packed A either in normal or
               transpose Format. The LDA of RFP A is (N+1)/2 when
               TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
               even and is N when is odd.
               See the Note below for more details. Unchanged on exit.

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
               Before entry,  the leading  m by n part of the array  B must
               contain  the  right-hand  side  matrix  B,  and  on exit  is
               overwritten by the solution matrix  X.

    LDB

              LDB is INTEGER
               On entry, LDB specifies the first dimension of B as declared
               in  the  calling  (sub)  program.   LDB  must  be  at  least
               max( 1, m ).
               Unchanged on exit.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 275 of file dtfsm.f.

  subroutine dtftri (character transr, character uplo, character diag, integer n, double precision, dimension( 0: * ) a, integer info)

DTFTRI

Purpose:

     DTFTRI computes the inverse of a triangular matrix A stored in RFP
     format.
     This is a Level 3 BLAS version of the algorithm.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal TRANSR of RFP A is stored;
              = 'T':  The Transpose TRANSR of RFP A is stored.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (0:nt-1);
              nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
              Positive Definite matrix A in RFP format. RFP format is
              described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
              then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
              (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
              the transpose of RFP A as defined when
              TRANSR = 'N'. The contents of RFP A are defined by UPLO as
              follows: If UPLO = 'U' the RFP A contains the nt elements of
              upper packed A; If UPLO = 'L' the RFP A contains the nt
              elements of lower packed A. The LDA of RFP A is (N+1)/2 when
              TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
              even and N is odd. See the Note below for more details.
              On exit, the (triangular) inverse of the original matrix, in
              the same storage format.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
                   matrix is singular and its inverse can not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 200 of file dtftri.f.

  subroutine dtfttp (character transr, character uplo, integer n, double precision, dimension( 0: * ) arf, double precision, dimension( 0: * ) ap, integer info)

DTFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).

Purpose:

     DTFTTP copies a triangular matrix A from rectangular full packed
     format (TF) to standard packed format (TP).

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  ARF is in Normal format;
              = 'T':  ARF is in Transpose format;

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    N

              N is INTEGER
              The order of the matrix A. N >= 0.

    ARF

              ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
              On entry, the upper or lower triangular matrix A stored in
              RFP format. For a further discussion see Notes below.

    AP

              AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
              On exit, the upper or lower triangular matrix A, packed
              columnwise in a linear array. The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 186 of file dtfttp.f.

  subroutine dtfttr (character transr, character uplo, integer n, double precision, dimension( 0: * ) arf, double precision, dimension( 0: lda-1, 0: * ) a, integer lda, integer info)

DTFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).

Purpose:

     DTFTTR copies a triangular matrix A from rectangular full packed
     format (TF) to standard full format (TR).

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  ARF is in Normal format;
              = 'T':  ARF is in Transpose format.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    N

              N is INTEGER
              The order of the matrices ARF and A. N >= 0.

    ARF

              ARF is DOUBLE PRECISION array, dimension (N*(N+1)/2).
              On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
              matrix A in RFP format. See the 'Notes' below for more
              details.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On exit, the triangular matrix A.  If UPLO = 'U', the
              leading N-by-N upper triangular part of the array A contains
              the upper triangular matrix, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of the array A contains
              the lower triangular matrix, and the strictly upper
              triangular part of A is not referenced.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 195 of file dtfttr.f.

  subroutine dtgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, integer m, double precision pl, double precision pr, double precision, dimension( * ) dif, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DTGSEN

Purpose:

     DTGSEN reorders the generalized real Schur decomposition of a real
     matrix pair (A, B) (in terms of an orthonormal equivalence trans-
     formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
     appears in the leading diagonal blocks of the upper quasi-triangular
     matrix A and the upper triangular B. The leading columns of Q and
     Z form orthonormal bases of the corresponding left and right eigen-
     spaces (deflating subspaces). (A, B) must be in generalized real
     Schur canonical form (as returned by DGGES), i.e. A is block upper
     triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
     triangular.
     DTGSEN also computes the generalized eigenvalues
                 w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
     of the reordered matrix pair (A, B).
     Optionally, DTGSEN computes the estimates of reciprocal condition
     numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
     (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
     between the matrix pairs (A11, B11) and (A22,B22) that correspond to
     the selected cluster and the eigenvalues outside the cluster, resp.,
     and norms of 'projections' onto left and right eigenspaces w.r.t.
     the selected cluster in the (1,1)-block.

Parameters
    IJOB

              IJOB is INTEGER
              Specifies whether condition numbers are required for the
              cluster of eigenvalues (PL and PR) or the deflating subspaces
              (Difu and Difl):
               =0: Only reorder w.r.t. SELECT. No extras.
               =1: Reciprocal of norms of 'projections' onto left and right
                   eigenspaces w.r.t. the selected cluster (PL and PR).
               =2: Upper bounds on Difu and Difl. F-norm-based estimate
                   (DIF(1:2)).
               =3: Estimate of Difu and Difl. 1-norm-based estimate
                   (DIF(1:2)).
                   About 5 times as expensive as IJOB = 2.
               =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
                   version to get it all.
               =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

    WANTQ

              WANTQ is LOGICAL
              .TRUE. : update the left transformation matrix Q;
              .FALSE.: do not update Q.

    WANTZ

              WANTZ is LOGICAL
              .TRUE. : update the right transformation matrix Z;
              .FALSE.: do not update Z.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              SELECT specifies the eigenvalues in the selected cluster.
              To select a real eigenvalue w(j), SELECT(j) must be set to
              .TRUE.. To select a complex conjugate pair of eigenvalues
              w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
              either SELECT(j) or SELECT(j+1) or both must be set to
              .TRUE.; a complex conjugate pair of eigenvalues must be
              either both included in the cluster or both excluded.

    N

              N is INTEGER
              The order of the matrices A and B. N >= 0.

    A

              A is DOUBLE PRECISION array, dimension(LDA,N)
              On entry, the upper quasi-triangular matrix A, with (A, B) in
              generalized real Schur canonical form.
              On exit, A is overwritten by the reordered matrix A.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension(LDB,N)
              On entry, the upper triangular matrix B, with (A, B) in
              generalized real Schur canonical form.
              On exit, B is overwritten by the reordered matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,N).

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
              be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
              and BETA(j),j=1,...,N  are the diagonals of the complex Schur
              form (S,T) that would result if the 2-by-2 diagonal blocks of
              the real generalized Schur form of (A,B) were further reduced
              to triangular form using complex unitary transformations.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) negative.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
              On exit, Q has been postmultiplied by the left orthogonal
              transformation matrix which reorder (A, B); The leading M
              columns of Q form orthonormal bases for the specified pair of
              left eigenspaces (deflating subspaces).
              If WANTQ = .FALSE., Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= 1;
              and if WANTQ = .TRUE., LDQ >= N.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ,N)
              On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
              On exit, Z has been postmultiplied by the left orthogonal
              transformation matrix which reorder (A, B); The leading M
              columns of Z form orthonormal bases for the specified pair of
              left eigenspaces (deflating subspaces).
              If WANTZ = .FALSE., Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z. LDZ >= 1;
              If WANTZ = .TRUE., LDZ >= N.

    M

              M is INTEGER
              The dimension of the specified pair of left and right eigen-
              spaces (deflating subspaces). 0 <= M <= N.

    PL

              PL is DOUBLE PRECISION

    PR

              PR is DOUBLE PRECISION
              If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
              reciprocal of the norm of 'projections' onto left and right
              eigenspaces with respect to the selected cluster.
              0 < PL, PR <= 1.
              If M = 0 or M = N, PL = PR  = 1.
              If IJOB = 0, 2 or 3, PL and PR are not referenced.

    DIF

              DIF is DOUBLE PRECISION array, dimension (2).
              If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
              If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
              Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
              estimates of Difu and Difl.
              If M = 0 or N, DIF(1:2) = F-norm([A, B]).
              If IJOB = 0 or 1, DIF is not referenced.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >=  4*N+16.
              If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
              If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK. LIWORK >= 1.
              If IJOB = 1, 2 or 4, LIWORK >=  N+6.
              If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
                =0: Successful exit.
                <0: If INFO = -i, the i-th argument had an illegal value.
                =1: Reordering of (A, B) failed because the transformed
                    matrix pair (A, B) would be too far from generalized
                    Schur form; the problem is very ill-conditioned.
                    (A, B) may have been partially reordered.
                    If requested, 0 is returned in DIF(*), PL and PR.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      DTGSEN first collects the selected eigenvalues by computing
      orthogonal U and W that move them to the top left corner of (A, B).
      In other words, the selected eigenvalues are the eigenvalues of
      (A11, B11) in:
                  U**T*(A, B)*W = (A11 A12) (B11 B12) n1
                                  ( 0  A22),( 0  B22) n2
                                    n1  n2    n1  n2
      where N = n1+n2 and U**T means the transpose of U. The first n1 columns
      of U and W span the specified pair of left and right eigenspaces
      (deflating subspaces) of (A, B).
      If (A, B) has been obtained from the generalized real Schur
      decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
      reordered generalized real Schur form of (C, D) is given by
               (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
      and the first n1 columns of Q*U and Z*W span the corresponding
      deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
      Note that if the selected eigenvalue is sufficiently ill-conditioned,
      then its value may differ significantly from its value before
      reordering.
      The reciprocal condition numbers of the left and right eigenspaces
      spanned by the first n1 columns of U and W (or Q*U and Z*W) may
      be returned in DIF(1:2), corresponding to Difu and Difl, resp.
      The Difu and Difl are defined as:
           Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
      and
           Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
      where sigma-min(Zu) is the smallest singular value of the
      (2*n1*n2)-by-(2*n1*n2) matrix
           Zu = [ kron(In2, A11)  -kron(A22**T, In1) ]
                [ kron(In2, B11)  -kron(B22**T, In1) ].
      Here, Inx is the identity matrix of size nx and A22**T is the
      transpose of A22. kron(X, Y) is the Kronecker product between
      the matrices X and Y.
      When DIF(2) is small, small changes in (A, B) can cause large changes
      in the deflating subspace. An approximate (asymptotic) bound on the
      maximum angular error in the computed deflating subspaces is
           EPS * norm((A, B)) / DIF(2),
      where EPS is the machine precision.
      The reciprocal norm of the projectors on the left and right
      eigenspaces associated with (A11, B11) may be returned in PL and PR.
      They are computed as follows. First we compute L and R so that
      P*(A, B)*Q is block diagonal, where
           P = ( I -L ) n1           Q = ( I R ) n1
               ( 0  I ) n2    and        ( 0 I ) n2
                 n1 n2                    n1 n2
      and (L, R) is the solution to the generalized Sylvester equation
           A11*R - L*A22 = -A12
           B11*R - L*B22 = -B12
      Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
      An approximate (asymptotic) bound on the average absolute error of
      the selected eigenvalues is
           EPS * norm((A, B)) / PL.
      There are also global error bounds which valid for perturbations up
      to a certain restriction:  A lower bound (x) on the smallest
      F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
      coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
      (i.e. (A + E, B + F), is
       x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
      An approximate bound on x can be computed from DIF(1:2), PL and PR.
      If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
      (L', R') and unperturbed (L, R) left and right deflating subspaces
      associated with the selected cluster in the (1,1)-blocks can be
      bounded as
       max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
       max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
      See LAPACK User's Guide section 4.11 or the following references
      for more information.
      Note that if the default method for computing the Frobenius-norm-
      based estimate DIF is not wanted (see DLATDF), then the parameter
      IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
      (IJOB = 2 will be used)). See DTGSYL for more details.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

      [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
          Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
          M.S. Moonen et al (eds), Linear Algebra for Large Scale and
          Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
      [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
          Eigenvalues of a Regular Matrix Pair (A, B) and Condition
          Estimation: Theory, Algorithms and Software,
          Report UMINF - 94.04, Department of Computing Science, Umea
          University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
          Note 87. To appear in Numerical Algorithms, 1996.
      [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
          for Solving the Generalized Sylvester Equation and Estimating the
          Separation between Regular Matrix Pairs, Report UMINF - 93.23,
          Department of Computing Science, Umea University, S-901 87 Umea,
          Sweden, December 1993, Revised April 1994, Also as LAPACK Working
          Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
          1996.

Definition at line 448 of file dtgsen.f.

  subroutine dtgsja (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer ncycle, integer info)

DTGSJA

Purpose:

     DTGSJA computes the generalized singular value decomposition (GSVD)
     of two real upper triangular (or trapezoidal) matrices A and B.
     On entry, it is assumed that matrices A and B have the following
     forms, which may be obtained by the preprocessing subroutine DGGSVP
     from a general M-by-N matrix A and P-by-N matrix B:
                  N-K-L  K    L
        A =    K ( 0    A12  A13 ) if M-K-L >= 0;
               L ( 0     0   A23 )
           M-K-L ( 0     0    0  )
                N-K-L  K    L
        A =  K ( 0    A12  A13 ) if M-K-L < 0;
           M-K ( 0     0   A23 )
                N-K-L  K    L
        B =  L ( 0     0   B13 )
           P-L ( 0     0    0  )
     where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
     upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
     otherwise A23 is (M-K)-by-L upper trapezoidal.
     On exit,
            U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),
     where U, V and Q are orthogonal matrices.
     R is a nonsingular upper triangular matrix, and D1 and D2 are
     ``diagonal'' matrices, which are of the following structures:
     If M-K-L >= 0,
                         K  L
            D1 =     K ( I  0 )
                     L ( 0  C )
                 M-K-L ( 0  0 )
                       K  L
            D2 = L   ( 0  S )
                 P-L ( 0  0 )
                    N-K-L  K    L
       ( 0 R ) = K (  0   R11  R12 ) K
                 L (  0    0   R22 ) L
     where
       C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
       S = diag( BETA(K+1),  ... , BETA(K+L) ),
       C**2 + S**2 = I.
       R is stored in A(1:K+L,N-K-L+1:N) on exit.
     If M-K-L < 0,
                    K M-K K+L-M
         D1 =   K ( I  0    0   )
              M-K ( 0  C    0   )
                      K M-K K+L-M
         D2 =   M-K ( 0  S    0   )
              K+L-M ( 0  0    I   )
                P-L ( 0  0    0   )
                    N-K-L  K   M-K  K+L-M
     ( 0 R ) =    K ( 0    R11  R12  R13  )
               M-K ( 0     0   R22  R23  )
             K+L-M ( 0     0    0   R33  )
     where
     C = diag( ALPHA(K+1), ... , ALPHA(M) ),
     S = diag( BETA(K+1),  ... , BETA(M) ),
     C**2 + S**2 = I.
     R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
         (  0  R22 R23 )
     in B(M-K+1:L,N+M-K-L+1:N) on exit.
     The computation of the orthogonal transformation matrices U, V or Q
     is optional.  These matrices may either be formed explicitly, or they
     may be postmultiplied into input matrices U1, V1, or Q1.

Parameters
    JOBU

              JOBU is CHARACTER*1
              = 'U':  U must contain an orthogonal matrix U1 on entry, and
                      the product U1*U is returned;
              = 'I':  U is initialized to the unit matrix, and the
                      orthogonal matrix U is returned;
              = 'N':  U is not computed.

    JOBV

              JOBV is CHARACTER*1
              = 'V':  V must contain an orthogonal matrix V1 on entry, and
                      the product V1*V is returned;
              = 'I':  V is initialized to the unit matrix, and the
                      orthogonal matrix V is returned;
              = 'N':  V is not computed.

    JOBQ

              JOBQ is CHARACTER*1
              = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
                      the product Q1*Q is returned;
              = 'I':  Q is initialized to the unit matrix, and the
                      orthogonal matrix Q is returned;
              = 'N':  Q is not computed.

    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    P

              P is INTEGER
              The number of rows of the matrix B.  P >= 0.

    N

              N is INTEGER
              The number of columns of the matrices A and B.  N >= 0.

    K

              K is INTEGER

    L

              L is INTEGER
              K and L specify the subblocks in the input matrices A and B:
              A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
              of A and B, whose GSVD is going to be computed by DTGSJA.
              See Further Details.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
              matrix R or part of R.  See Purpose for details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
              a part of R.  See Purpose for details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,P).

    TOLA

              TOLA is DOUBLE PRECISION

    TOLB

              TOLB is DOUBLE PRECISION
              TOLA and TOLB are the convergence criteria for the Jacobi-
              Kogbetliantz iteration procedure. Generally, they are the
              same as used in the preprocessing step, say
                  TOLA = max(M,N)*norm(A)*MAZHEPS,
                  TOLB = max(P,N)*norm(B)*MAZHEPS.

    ALPHA

              ALPHA is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, ALPHA and BETA contain the generalized singular
              value pairs of A and B;
                ALPHA(1:K) = 1,
                BETA(1:K)  = 0,
              and if M-K-L >= 0,
                ALPHA(K+1:K+L) = diag(C),
                BETA(K+1:K+L)  = diag(S),
              or if M-K-L < 0,
                ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
                BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
              Furthermore, if K+L < N,
                ALPHA(K+L+1:N) = 0 and
                BETA(K+L+1:N)  = 0.

    U

              U is DOUBLE PRECISION array, dimension (LDU,M)
              On entry, if JOBU = 'U', U must contain a matrix U1 (usually
              the orthogonal matrix returned by DGGSVP).
              On exit,
              if JOBU = 'I', U contains the orthogonal matrix U;
              if JOBU = 'U', U contains the product U1*U.
              If JOBU = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U. LDU >= max(1,M) if
              JOBU = 'U'; LDU >= 1 otherwise.

    V

              V is DOUBLE PRECISION array, dimension (LDV,P)
              On entry, if JOBV = 'V', V must contain a matrix V1 (usually
              the orthogonal matrix returned by DGGSVP).
              On exit,
              if JOBV = 'I', V contains the orthogonal matrix V;
              if JOBV = 'V', V contains the product V1*V.
              If JOBV = 'N', V is not referenced.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,P) if
              JOBV = 'V'; LDV >= 1 otherwise.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
              the orthogonal matrix returned by DGGSVP).
              On exit,
              if JOBQ = 'I', Q contains the orthogonal matrix Q;
              if JOBQ = 'Q', Q contains the product Q1*Q.
              If JOBQ = 'N', Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q. LDQ >= max(1,N) if
              JOBQ = 'Q'; LDQ >= 1 otherwise.

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)

    NCYCLE

              NCYCLE is INTEGER
              The number of cycles required for convergence.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              = 1:  the procedure does not converge after MAXIT cycles.

  Internal Parameters
  ===================
  MAXIT   INTEGER
          MAXIT specifies the total loops that the iterative procedure
          may take. If after MAXIT cycles, the routine fails to
          converge, we return INFO = 1..fi

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
      min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
      matrix B13 to the form:
               U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
      where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
      of Z.  C1 and S1 are diagonal matrices satisfying
                    C1**2 + S1**2 = I,
      and R1 is an L-by-L nonsingular upper triangular matrix.

Definition at line 375 of file dtgsja.f.

  subroutine dtgsna (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) dif, integer mm, integer m, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DTGSNA

Purpose:

     DTGSNA estimates reciprocal condition numbers for specified
     eigenvalues and/or eigenvectors of a matrix pair (A, B) in
     generalized real Schur canonical form (or of any matrix pair
     (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
     Z**T denotes the transpose of Z.
     (A, B) must be in generalized real Schur form (as returned by DGGES),
     i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
     blocks. B is upper triangular.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies whether condition numbers are required for
              eigenvalues (S) or eigenvectors (DIF):
              = 'E': for eigenvalues only (S);
              = 'V': for eigenvectors only (DIF);
              = 'B': for both eigenvalues and eigenvectors (S and DIF).

    HOWMNY

              HOWMNY is CHARACTER*1
              = 'A': compute condition numbers for all eigenpairs;
              = 'S': compute condition numbers for selected eigenpairs
                     specified by the array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY = 'S', SELECT specifies the eigenpairs for which
              condition numbers are required. To select condition numbers
              for the eigenpair corresponding to a real eigenvalue w(j),
              SELECT(j) must be set to .TRUE.. To select condition numbers
              corresponding to a complex conjugate pair of eigenvalues w(j)
              and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
              set to .TRUE..
              If HOWMNY = 'A', SELECT is not referenced.

    N

              N is INTEGER
              The order of the square matrix pair (A, B). N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              The upper quasi-triangular matrix A in the pair (A,B).

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              The upper triangular matrix B in the pair (A,B).

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,N).

    VL

              VL is DOUBLE PRECISION array, dimension (LDVL,M)
              If JOB = 'E' or 'B', VL must contain left eigenvectors of
              (A, B), corresponding to the eigenpairs specified by HOWMNY
              and SELECT. The eigenvectors must be stored in consecutive
              columns of VL, as returned by DTGEVC.
              If JOB = 'V', VL is not referenced.

    LDVL

              LDVL is INTEGER
              The leading dimension of the array VL. LDVL >= 1.
              If JOB = 'E' or 'B', LDVL >= N.

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,M)
              If JOB = 'E' or 'B', VR must contain right eigenvectors of
              (A, B), corresponding to the eigenpairs specified by HOWMNY
              and SELECT. The eigenvectors must be stored in consecutive
              columns ov VR, as returned by DTGEVC.
              If JOB = 'V', VR is not referenced.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR. LDVR >= 1.
              If JOB = 'E' or 'B', LDVR >= N.

    S

              S is DOUBLE PRECISION array, dimension (MM)
              If JOB = 'E' or 'B', the reciprocal condition numbers of the
              selected eigenvalues, stored in consecutive elements of the
              array. For a complex conjugate pair of eigenvalues two
              consecutive elements of S are set to the same value. Thus
              S(j), DIF(j), and the j-th columns of VL and VR all
              correspond to the same eigenpair (but not in general the
              j-th eigenpair, unless all eigenpairs are selected).
              If JOB = 'V', S is not referenced.

    DIF

              DIF is DOUBLE PRECISION array, dimension (MM)
              If JOB = 'V' or 'B', the estimated reciprocal condition
              numbers of the selected eigenvectors, stored in consecutive
              elements of the array. For a complex eigenvector two
              consecutive elements of DIF are set to the same value. If
              the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
              is set to 0; this can only occur when the true value would be
              very small anyway.
              If JOB = 'E', DIF is not referenced.

    MM

              MM is INTEGER
              The number of elements in the arrays S and DIF. MM >= M.

    M

              M is INTEGER
              The number of elements of the arrays S and DIF used to store
              the specified condition numbers; for each selected real
              eigenvalue one element is used, and for each selected complex
              conjugate pair of eigenvalues, two elements are used.
              If HOWMNY = 'A', M is set to N.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N).
              If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (N + 6)
              If JOB = 'E', IWORK is not referenced.

    INFO

              INFO is INTEGER
              =0: Successful exit
              <0: If INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The reciprocal of the condition number of a generalized eigenvalue
      w = (a, b) is defined as
           S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
      where u and v are the left and right eigenvectors of (A, B)
      corresponding to w; |z| denotes the absolute value of the complex
      number, and norm(u) denotes the 2-norm of the vector u.
      The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
      of the matrix pair (A, B). If both a and b equal zero, then (A B) is
      singular and S(I) = -1 is returned.
      An approximate error bound on the chordal distance between the i-th
      computed generalized eigenvalue w and the corresponding exact
      eigenvalue lambda is
           chord(w, lambda) <= EPS * norm(A, B) / S(I)
      where EPS is the machine precision.
      The reciprocal of the condition number DIF(i) of right eigenvector u
      and left eigenvector v corresponding to the generalized eigenvalue w
      is defined as follows:
      a) If the i-th eigenvalue w = (a,b) is real
         Suppose U and V are orthogonal transformations such that
                  U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
                                            ( 0  S22 ),( 0 T22 )  n-1
                                              1  n-1     1 n-1
         Then the reciprocal condition number DIF(i) is
                    Difl((a, b), (S22, T22)) = sigma-min( Zl ),
         where sigma-min(Zl) denotes the smallest singular value of the
         2(n-1)-by-2(n-1) matrix
             Zl = [ kron(a, In-1)  -kron(1, S22) ]
                  [ kron(b, In-1)  -kron(1, T22) ] .
         Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
         Kronecker product between the matrices X and Y.
         Note that if the default method for computing DIF(i) is wanted
         (see DLATDF), then the parameter DIFDRI (see below) should be
         changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
         See DTGSYL for more details.
      b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
         Suppose U and V are orthogonal transformations such that
                  U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
                                           ( 0    S22 ),( 0    T22) n-2
                                             2    n-2     2    n-2
         and (S11, T11) corresponds to the complex conjugate eigenvalue
         pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
         that
           U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
                          (  0  s22 )                    (  0  t22 )
         where the generalized eigenvalues w = s11/t11 and
         conjg(w) = s22/t22.
         Then the reciprocal condition number DIF(i) is bounded by
             min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
         where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
         Z1 is the complex 2-by-2 matrix
                  Z1 =  [ s11  -s22 ]
                        [ t11  -t22 ],
         This is done by computing (using real arithmetic) the
         roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
         where Z1**T denotes the transpose of Z1 and det(X) denotes
         the determinant of X.
         and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
         upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
                  Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
                       [ kron(T11**T, In-2)  -kron(I2, T22) ]
         Note that if the default method for computing DIF is wanted (see
         DLATDF), then the parameter DIFDRI (see below) should be changed
         from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
         for more details.
      For each eigenvalue/vector specified by SELECT, DIF stores a
      Frobenius norm-based estimate of Difl.
      An approximate error bound for the i-th computed eigenvector VL(i) or
      VR(i) is given by
                 EPS * norm(A, B) / DIF(i).
      See ref. [2-3] for more details and further references.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

      [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
          Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
          M.S. Moonen et al (eds), Linear Algebra for Large Scale and
          Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
      [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
          Eigenvalues of a Regular Matrix Pair (A, B) and Condition
          Estimation: Theory, Algorithms and Software,
          Report UMINF - 94.04, Department of Computing Science, Umea
          University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
          Note 87. To appear in Numerical Algorithms, 1996.
      [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
          for Solving the Generalized Sylvester Equation and Estimating the
          Separation between Regular Matrix Pairs, Report UMINF - 93.23,
          Department of Computing Science, Umea University, S-901 87 Umea,
          Sweden, December 1993, Revised April 1994, Also as LAPACK Working
          Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
          No 1, 1996.

Definition at line 378 of file dtgsna.f.

  subroutine dtpcon (character norm, character uplo, character diag, integer n, double precision, dimension( * ) ap, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DTPCON

Purpose:

     DTPCON estimates the reciprocal of the condition number of a packed
     triangular matrix A, in either the 1-norm or the infinity-norm.
     The norm of A is computed and an estimate is obtained for
     norm(inv(A)), then the reciprocal of the condition number is
     computed as
        RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies whether the 1-norm condition number or the
              infinity-norm condition number is required:
              = '1' or 'O':  1-norm;
              = 'I':         Infinity-norm.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangular matrix A, packed columnwise in
              a linear array.  The j-th column of A is stored in the array
              AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              If DIAG = 'U', the diagonal elements of A are not referenced
              and are assumed to be 1.

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 128 of file dtpcon.f.

  subroutine dtplqt (integer m, integer n, integer l, integer mb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer info)

DTPLQT

Purpose:

     DTPLQT computes a blocked LQ factorization of a real
     'triangular-pentagonal' matrix C, which is composed of a
     triangular block A and pentagonal block B, using the compact
     WY representation for Q.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix B, and the order of the
              triangular matrix A.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size to be used in the blocked QR.  M >= MB >= 1.

    A

              A is DOUBLE PRECISION array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The lower triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.  See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MB*M)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ] [ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
      upper trapezoidal matrix B2:
              [ B ] = [ B1 ] [ B2 ]
                       [ B1 ]  <- M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                [ C ] = [ A ] [ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                [ W ] = [ I ] [ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                [ V ] = [ V1 ] [ V2 ]
                       [ V1 ] <- M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The number of blocks is B = ceiling(M/MB), where each
      block is of order MB except for the last block, which is of order
      IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-N matrix T as
                   T = [T1 T2 ... TB].

Definition at line 187 of file dtplqt.f.

  subroutine dtplqt2 (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, integer info)

DTPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

     DTPLQT2 computes a LQ a factorization of a real 'triangular-pentagonal'
     matrix C, which is composed of a triangular block A and pentagonal block B,
     using the compact WY representation for Q.

Parameters
    M

              M is INTEGER
              The total number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B, and the order of
              the triangular matrix A.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    A

              A is DOUBLE PRECISION array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is DOUBLE PRECISION array, dimension (LDT,M)
              The N-by-N upper triangular factor T of the block reflector.
              See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= max(1,M)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ][ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
      upper trapezoidal matrix B2:
                   B = [ B1 ][ B2 ]
                       [ B1 ]  <-     M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                   C = [ A ][ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                   W = [ I ][ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                   W = [ V1 ][ V2 ]
                       [ V1 ] <-     M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The (M+N)-by-(M+N) block reflector H is then given by
                   H = I - W**T * T * W
      where W^H is the conjugate transpose of W and T is the upper triangular
      factor of the block reflector.

Definition at line 176 of file dtplqt2.f.

  subroutine dtpmlqt (character side, character trans, integer m, integer n, integer k, integer l, integer mb, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) work, integer info)

DTPMLQT

Purpose:

     DTPMQRT applies a real orthogonal matrix Q obtained from a
     'triangular-pentagonal' real block reflector H to a general
     real matrix C, which consists of two blocks A and B.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix B. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.

    L

              L is INTEGER
              The order of the trapezoidal part of V.
              K >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size used for the storage of T.  K >= MB >= 1.
              This must be the same value of MB used to generate T
              in DTPLQT.

    V

              V is DOUBLE PRECISION array, dimension (LDV,K)
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DTPLQT in B.  See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= K.

    T

              T is DOUBLE PRECISION array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by DTPLQT, stored as a MB-by-K matrix.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    A

              A is DOUBLE PRECISION array, dimension
              (LDA,N) if SIDE = 'L' or
              (LDA,K) if SIDE = 'R'
              On entry, the K-by-N or M-by-K matrix A.
              On exit, A is overwritten by the corresponding block of
              Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,K);
              If SIDE = 'R', LDA >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the M-by-N matrix B.
              On exit, B is overwritten by the corresponding block of
              Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.
              LDB >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array. The dimension of WORK is
               N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The columns of the pentagonal matrix V contain the elementary reflectors
      H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
      trapezoidal block V2:
            V = [V1] [V2].
      The size of the trapezoidal block V2 is determined by the parameter L,
      where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L
      rows of a K-by-K upper triangular matrix.  If L=K, V2 is lower triangular;
      if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
      If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                          [B]
      If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.
      The real orthogonal matrix Q is formed from V and T.
      If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
      If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
      If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
      If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.

Definition at line 212 of file dtpmlqt.f.

  subroutine dtpmqrt (character side, character trans, integer m, integer n, integer k, integer l, integer nb, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) work, integer info)

DTPMQRT

Purpose:

     DTPMQRT applies a real orthogonal matrix Q obtained from a
     'triangular-pentagonal' real block reflector H to a general
     real matrix C, which consists of two blocks A and B.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix B. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.

    L

              L is INTEGER
              The order of the trapezoidal part of V.
              K >= L >= 0.  See Further Details.

    NB

              NB is INTEGER
              The block size used for the storage of T.  K >= NB >= 1.
              This must be the same value of NB used to generate T
              in CTPQRT.

    V

              V is DOUBLE PRECISION array, dimension (LDV,K)
              The i-th column must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              CTPQRT in B.  See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If SIDE = 'L', LDV >= max(1,M);
              if SIDE = 'R', LDV >= max(1,N).

    T

              T is DOUBLE PRECISION array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by CTPQRT, stored as a NB-by-K matrix.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= NB.

    A

              A is DOUBLE PRECISION array, dimension
              (LDA,N) if SIDE = 'L' or
              (LDA,K) if SIDE = 'R'
              On entry, the K-by-N or M-by-K matrix A.
              On exit, A is overwritten by the corresponding block of
              Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDC >= max(1,K);
              If SIDE = 'R', LDC >= max(1,M).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the M-by-N matrix B.
              On exit, B is overwritten by the corresponding block of
              Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.
              LDB >= max(1,M).

    WORK

              WORK is DOUBLE PRECISION array. The dimension of WORK is
               N*NB if SIDE = 'L', or  M*NB if SIDE = 'R'.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The columns of the pentagonal matrix V contain the elementary reflectors
      H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
      trapezoidal block V2:
            V = [V1]
                [V2].
      The size of the trapezoidal block V2 is determined by the parameter L,
      where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L
      rows of a K-by-K upper triangular matrix.  If L=K, V2 is upper triangular;
      if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
      If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is M-by-K.
                          [B]
      If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is N-by-K.
      The real orthogonal matrix Q is formed from V and T.
      If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
      If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
      If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
      If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.

Definition at line 214 of file dtpmqrt.f.

  subroutine dtpqrt (integer m, integer n, integer l, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer info)

DTPQRT

Purpose:

     DTPQRT computes a blocked QR factorization of a real
     'triangular-pentagonal' matrix C, which is composed of a
     triangular block A and pentagonal block B, using the compact
     WY representation for Q.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B, and the order of the
              triangular matrix A.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the upper trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    NB

              NB is INTEGER
              The block size to be used in the blocked QR.  N >= NB >= 1.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the upper triangular N-by-N matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the upper triangular matrix R.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first M-L rows
              are rectangular, and the last L rows are upper trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.  See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= NB.

    WORK

              WORK is DOUBLE PRECISION array, dimension (NB*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a (N+M)-by-N matrix
                   C = [ A ]
                       [ B ]
      where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
      matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
      upper trapezoidal matrix B2:
                   B = [ B1 ]  <- (M-L)-by-N rectangular
                       [ B2 ]  <-     L-by-N upper trapezoidal.
      The upper trapezoidal matrix B2 consists of the first L rows of a
      N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is upper triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th column
      below the diagonal (of A) in the (N+M)-by-N input matrix C
                   C = [ A ]  <- upper triangular N-by-N
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                   W = [ I ]  <- identity, N-by-N
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                   V = [ V1 ] <- (M-L)-by-N rectangular
                       [ V2 ] <-     L-by-N upper trapezoidal.
      The columns of V represent the vectors which define the H(i)'s.
      The number of blocks is B = ceiling(N/NB), where each
      block is of order NB except for the last block, which is of order
      IB = N - (B-1)*NB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
      for the last block) T's are stored in the NB-by-N matrix T as
                   T = [T1 T2 ... TB].

Definition at line 187 of file dtpqrt.f.

  subroutine dtpqrt2 (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, integer info)

DTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

     DTPQRT2 computes a QR factorization of a real 'triangular-pentagonal'
     matrix C, which is composed of a triangular block A and pentagonal block B,
     using the compact WY representation for Q.

Parameters
    M

              M is INTEGER
              The total number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B, and the order of
              the triangular matrix A.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the upper trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the upper triangular N-by-N matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the upper triangular matrix R.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first M-L rows
              are rectangular, and the last L rows are upper trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The N-by-N upper triangular factor T of the block reflector.
              See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= max(1,N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a (N+M)-by-N matrix
                   C = [ A ]
                       [ B ]
      where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
      matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
      upper trapezoidal matrix B2:
                   B = [ B1 ]  <- (M-L)-by-N rectangular
                       [ B2 ]  <-     L-by-N upper trapezoidal.
      The upper trapezoidal matrix B2 consists of the first L rows of a
      N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is upper triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th column
      below the diagonal (of A) in the (N+M)-by-N input matrix C
                   C = [ A ]  <- upper triangular N-by-N
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                   W = [ I ]  <- identity, N-by-N
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                   V = [ V1 ] <- (M-L)-by-N rectangular
                       [ V2 ] <-     L-by-N upper trapezoidal.
      The columns of V represent the vectors which define the H(i)'s.
      The (M+N)-by-(M+N) block reflector H is then given by
                   H = I - W * T * W**T
      where W^H is the conjugate transpose of W and T is the upper triangular
      factor of the block reflector.

Definition at line 172 of file dtpqrt2.f.

  subroutine dtprfs (character uplo, character trans, character diag, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DTPRFS

Purpose:

     DTPRFS provides error bounds and backward error estimates for the
     solution to a system of linear equations with a triangular packed
     coefficient matrix.
     The solution matrix X must be computed by DTPTRS or some other
     means before entering this routine.  DTPRFS does not do iterative
     refinement because doing so cannot improve the backward error.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B  (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangular matrix A, packed columnwise in
              a linear array.  The j-th column of A is stored in the array
              AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              If DIAG = 'U', the diagonal elements of A are not referenced
              and are assumed to be 1.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              The solution matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 173 of file dtprfs.f.

  subroutine dtptri (character uplo, character diag, integer n, double precision, dimension( * ) ap, integer info)

DTPTRI

Purpose:

     DTPTRI computes the inverse of a real upper or lower triangular
     matrix A stored in packed format.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangular matrix A, stored
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.
              On exit, the (triangular) inverse of the original matrix, in
              the same packed storage format.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, A(i,i) is exactly zero.  The triangular
                    matrix is singular and its inverse can not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      A triangular matrix A can be transferred to packed storage using one
      of the following program segments:
      UPLO = 'U':                      UPLO = 'L':
            JC = 1                           JC = 1
            DO 2 J = 1, N                    DO 2 J = 1, N
               DO 1 I = 1, J                    DO 1 I = J, N
                  AP(JC+I-1) = A(I,J)              AP(JC+I-J) = A(I,J)
          1    CONTINUE                    1    CONTINUE
               JC = JC + J                      JC = JC + N - J + 1
          2 CONTINUE                       2 CONTINUE

Definition at line 116 of file dtptri.f.

  subroutine dtptrs (character uplo, character trans, character diag, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DTPTRS

Purpose:

     DTPTRS solves a triangular system of the form
        A * X = B  or  A**T * X = B,
     where A is a triangular matrix of order N stored in packed format,
     and B is an N-by-NRHS matrix.  A check is made to verify that A is
     nonsingular.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B  (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              The upper or lower triangular matrix A, packed columnwise in
              a linear array.  The j-th column of A is stored in the array
              AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, if INFO = 0, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the i-th diagonal element of A is zero,
                    indicating that the matrix is singular and the
                    solutions X have not been computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 129 of file dtptrs.f.

  subroutine dtpttf (character transr, character uplo, integer n, double precision, dimension( 0: * ) ap, double precision, dimension( 0: * ) arf, integer info)

DTPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).

Purpose:

     DTPTTF copies a triangular matrix A from standard packed format (TP)
     to rectangular full packed format (TF).

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  ARF in Normal format is wanted;
              = 'T':  ARF in Conjugate-transpose format is wanted.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
              On entry, the upper or lower triangular matrix A, packed
              columnwise in a linear array. The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    ARF

              ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
              On exit, the upper or lower triangular matrix A stored in
              RFP format. For a further discussion see Notes below.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 185 of file dtpttf.f.

  subroutine dtpttr (character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( lda, * ) a, integer lda, integer info)

DTPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).

Purpose:

     DTPTTR copies a triangular matrix A from standard packed format (TP)
     to standard full format (TR).

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular.
              = 'L':  A is lower triangular.

    N

              N is INTEGER
              The order of the matrix A. N >= 0.

    AP

              AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
              On entry, the upper or lower triangular matrix A, packed
              columnwise in a linear array. The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    A

              A is DOUBLE PRECISION array, dimension ( LDA, N )
              On exit, the triangular matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 103 of file dtpttr.f.

  subroutine dtrcon (character norm, character uplo, character diag, integer n, double precision, dimension( lda, * ) a, integer lda, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DTRCON

Purpose:

     DTRCON estimates the reciprocal of the condition number of a
     triangular matrix A, in either the 1-norm or the infinity-norm.
     The norm of A is computed and an estimate is obtained for
     norm(inv(A)), then the reciprocal of the condition number is
     computed as
        RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies whether the 1-norm condition number or the
              infinity-norm condition number is required:
              = '1' or 'O':  1-norm;
              = 'I':         Infinity-norm.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading N-by-N
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading N-by-N lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    RCOND

              RCOND is DOUBLE PRECISION
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 135 of file dtrcon.f.

  subroutine dtrevc (character side, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer info)

DTREVC

Purpose:

     DTREVC computes some or all of the right and/or left eigenvectors of
     a real upper quasi-triangular matrix T.
     Matrices of this type are produced by the Schur factorization of
     a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
     The right eigenvector x and the left eigenvector y of T corresponding
     to an eigenvalue w are defined by:
        T*x = w*x,     (y**H)*T = w*(y**H)
     where y**H denotes the conjugate transpose of y.
     The eigenvalues are not input to this routine, but are read directly
     from the diagonal blocks of T.
     This routine returns the matrices X and/or Y of right and left
     eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
     input matrix.  If Q is the orthogonal factor that reduces a matrix
     A to Schur form T, then Q*X and Q*Y are the matrices of right and
     left eigenvectors of A.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'R':  compute right eigenvectors only;
              = 'L':  compute left eigenvectors only;
              = 'B':  compute both right and left eigenvectors.

    HOWMNY

              HOWMNY is CHARACTER*1
              = 'A':  compute all right and/or left eigenvectors;
              = 'B':  compute all right and/or left eigenvectors,
                      backtransformed by the matrices in VR and/or VL;
              = 'S':  compute selected right and/or left eigenvectors,
                      as indicated by the logical array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY = 'S', SELECT specifies the eigenvectors to be
              computed.
              If w(j) is a real eigenvalue, the corresponding real
              eigenvector is computed if SELECT(j) is .TRUE..
              If w(j) and w(j+1) are the real and imaginary parts of a
              complex eigenvalue, the corresponding complex eigenvector is
              computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
              on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
              .FALSE..
              Not referenced if HOWMNY = 'A' or 'B'.

    N

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The upper quasi-triangular matrix T in Schur canonical form.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    VL

              VL is DOUBLE PRECISION array, dimension (LDVL,MM)
              On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
              contain an N-by-N matrix Q (usually the orthogonal matrix Q
              of Schur vectors returned by DHSEQR).
              On exit, if SIDE = 'L' or 'B', VL contains:
              if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
              if HOWMNY = 'B', the matrix Q*Y;
              if HOWMNY = 'S', the left eigenvectors of T specified by
                               SELECT, stored consecutively in the columns
                               of VL, in the same order as their
                               eigenvalues.
              A complex eigenvector corresponding to a complex eigenvalue
              is stored in two consecutive columns, the first holding the
              real part, and the second the imaginary part.
              Not referenced if SIDE = 'R'.

    LDVL

              LDVL is INTEGER
              The leading dimension of the array VL.  LDVL >= 1, and if
              SIDE = 'L' or 'B', LDVL >= N.

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,MM)
              On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
              contain an N-by-N matrix Q (usually the orthogonal matrix Q
              of Schur vectors returned by DHSEQR).
              On exit, if SIDE = 'R' or 'B', VR contains:
              if HOWMNY = 'A', the matrix X of right eigenvectors of T;
              if HOWMNY = 'B', the matrix Q*X;
              if HOWMNY = 'S', the right eigenvectors of T specified by
                               SELECT, stored consecutively in the columns
                               of VR, in the same order as their
                               eigenvalues.
              A complex eigenvector corresponding to a complex eigenvalue
              is stored in two consecutive columns, the first holding the
              real part and the second the imaginary part.
              Not referenced if SIDE = 'L'.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR.  LDVR >= 1, and if
              SIDE = 'R' or 'B', LDVR >= N.

    MM

              MM is INTEGER
              The number of columns in the arrays VL and/or VR. MM >= M.

    M

              M is INTEGER
              The number of columns in the arrays VL and/or VR actually
              used to store the eigenvectors.
              If HOWMNY = 'A' or 'B', M is set to N.
              Each selected real eigenvector occupies one column and each
              selected complex eigenvector occupies two columns.

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The algorithm used in this program is basically backward (forward)
      substitution, with scaling to make the the code robust against
      possible overflow.
      Each eigenvector is normalized so that the element of largest
      magnitude has magnitude 1; here the magnitude of a complex number
      (x,y) is taken to be |x| + |y|.

Definition at line 220 of file dtrevc.f.

  subroutine dtrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer lwork, integer info)

DTREVC3

Purpose:

     DTREVC3 computes some or all of the right and/or left eigenvectors of
     a real upper quasi-triangular matrix T.
     Matrices of this type are produced by the Schur factorization of
     a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.
     The right eigenvector x and the left eigenvector y of T corresponding
     to an eigenvalue w are defined by:
        T*x = w*x,     (y**T)*T = w*(y**T)
     where y**T denotes the transpose of the vector y.
     The eigenvalues are not input to this routine, but are read directly
     from the diagonal blocks of T.
     This routine returns the matrices X and/or Y of right and left
     eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
     input matrix. If Q is the orthogonal factor that reduces a matrix
     A to Schur form T, then Q*X and Q*Y are the matrices of right and
     left eigenvectors of A.
     This uses a Level 3 BLAS version of the back transformation.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'R':  compute right eigenvectors only;
              = 'L':  compute left eigenvectors only;
              = 'B':  compute both right and left eigenvectors.

    HOWMNY

              HOWMNY is CHARACTER*1
              = 'A':  compute all right and/or left eigenvectors;
              = 'B':  compute all right and/or left eigenvectors,
                      backtransformed by the matrices in VR and/or VL;
              = 'S':  compute selected right and/or left eigenvectors,
                      as indicated by the logical array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY = 'S', SELECT specifies the eigenvectors to be
              computed.
              If w(j) is a real eigenvalue, the corresponding real
              eigenvector is computed if SELECT(j) is .TRUE..
              If w(j) and w(j+1) are the real and imaginary parts of a
              complex eigenvalue, the corresponding complex eigenvector is
              computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
              on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
              .FALSE..
              Not referenced if HOWMNY = 'A' or 'B'.

    N

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The upper quasi-triangular matrix T in Schur canonical form.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    VL

              VL is DOUBLE PRECISION array, dimension (LDVL,MM)
              On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
              contain an N-by-N matrix Q (usually the orthogonal matrix Q
              of Schur vectors returned by DHSEQR).
              On exit, if SIDE = 'L' or 'B', VL contains:
              if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
              if HOWMNY = 'B', the matrix Q*Y;
              if HOWMNY = 'S', the left eigenvectors of T specified by
                               SELECT, stored consecutively in the columns
                               of VL, in the same order as their
                               eigenvalues.
              A complex eigenvector corresponding to a complex eigenvalue
              is stored in two consecutive columns, the first holding the
              real part, and the second the imaginary part.
              Not referenced if SIDE = 'R'.

    LDVL

              LDVL is INTEGER
              The leading dimension of the array VL.
              LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,MM)
              On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
              contain an N-by-N matrix Q (usually the orthogonal matrix Q
              of Schur vectors returned by DHSEQR).
              On exit, if SIDE = 'R' or 'B', VR contains:
              if HOWMNY = 'A', the matrix X of right eigenvectors of T;
              if HOWMNY = 'B', the matrix Q*X;
              if HOWMNY = 'S', the right eigenvectors of T specified by
                               SELECT, stored consecutively in the columns
                               of VR, in the same order as their
                               eigenvalues.
              A complex eigenvector corresponding to a complex eigenvalue
              is stored in two consecutive columns, the first holding the
              real part and the second the imaginary part.
              Not referenced if SIDE = 'L'.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR.
              LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.

    MM

              MM is INTEGER
              The number of columns in the arrays VL and/or VR. MM >= M.

    M

              M is INTEGER
              The number of columns in the arrays VL and/or VR actually
              used to store the eigenvectors.
              If HOWMNY = 'A' or 'B', M is set to N.
              Each selected real eigenvector occupies one column and each
              selected complex eigenvector occupies two columns.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))

    LWORK

              LWORK is INTEGER
              The dimension of array WORK. LWORK >= max(1,3*N).
              For optimum performance, LWORK >= N + 2*N*NB, where NB is
              the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The algorithm used in this program is basically backward (forward)
      substitution, with scaling to make the the code robust against
      possible overflow.
      Each eigenvector is normalized so that the element of largest
      magnitude has magnitude 1; here the magnitude of a complex number
      (x,y) is taken to be |x| + |y|.

Definition at line 235 of file dtrevc3.f.

  subroutine dtrexc (character compq, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldq, * ) q, integer ldq, integer ifst, integer ilst, double precision, dimension( * ) work, integer info)

DTREXC

Purpose:

     DTREXC reorders the real Schur factorization of a real matrix
     A = Q*T*Q**T, so that the diagonal block of T with row index IFST is
     moved to row ILST.
     The real Schur form T is reordered by an orthogonal similarity
     transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors
     is updated by postmultiplying it with Z.
     T must be in Schur canonical form (as returned by DHSEQR), that is,
     block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
     2-by-2 diagonal block has its diagonal elements equal and its
     off-diagonal elements of opposite sign.

Parameters
    COMPQ

              COMPQ is CHARACTER*1
              = 'V':  update the matrix Q of Schur vectors;
              = 'N':  do not update Q.

    N

              N is INTEGER
              The order of the matrix T. N >= 0.
              If N == 0 arguments ILST and IFST may be any value.

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              On entry, the upper quasi-triangular matrix T, in Schur
              Schur canonical form.
              On exit, the reordered upper quasi-triangular matrix, again
              in Schur canonical form.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
              On exit, if COMPQ = 'V', Q has been postmultiplied by the
              orthogonal transformation matrix Z which reorders T.
              If COMPQ = 'N', Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= 1, and if
              COMPQ = 'V', LDQ >= max(1,N).

    IFST

              IFST is INTEGER

    ILST

              ILST is INTEGER
              Specify the reordering of the diagonal blocks of T.
              The block with row index IFST is moved to row ILST, by a
              sequence of transpositions between adjacent blocks.
              On exit, if IFST pointed on entry to the second row of a
              2-by-2 block, it is changed to point to the first row; ILST
              always points to the first row of the block in its final
              position (which may differ from its input value by +1 or -1).
              1 <= IFST <= N; 1 <= ILST <= N.

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              = 1:  two adjacent blocks were too close to swap (the problem
                    is very ill-conditioned); T may have been partially
                    reordered, and ILST points to the first row of the
                    current position of the block being moved.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 146 of file dtrexc.f.

  subroutine dtrrfs (character uplo, character trans, character diag, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DTRRFS

Purpose:

     DTRRFS provides error bounds and backward error estimates for the
     solution to a system of linear equations with a triangular
     coefficient matrix.
     The solution matrix X must be computed by DTRTRS or some other
     means before entering this routine.  DTRRFS does not do iterative
     refinement because doing so cannot improve the backward error.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B  (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading N-by-N
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading N-by-N lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              The solution matrix X.

    LDX

              LDX is INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 180 of file dtrrfs.f.

  subroutine dtrsen (character job, character compq, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) wr, double precision, dimension( * ) wi, integer m, double precision s, double precision sep, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DTRSEN

Purpose:

     DTRSEN reorders the real Schur factorization of a real matrix
     A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
     the leading diagonal blocks of the upper quasi-triangular matrix T,
     and the leading columns of Q form an orthonormal basis of the
     corresponding right invariant subspace.
     Optionally the routine computes the reciprocal condition numbers of
     the cluster of eigenvalues and/or the invariant subspace.
     T must be in Schur canonical form (as returned by DHSEQR), that is,
     block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
     2-by-2 diagonal block has its diagonal elements equal and its
     off-diagonal elements of opposite sign.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies whether condition numbers are required for the
              cluster of eigenvalues (S) or the invariant subspace (SEP):
              = 'N': none;
              = 'E': for eigenvalues only (S);
              = 'V': for invariant subspace only (SEP);
              = 'B': for both eigenvalues and invariant subspace (S and
                     SEP).

    COMPQ

              COMPQ is CHARACTER*1
              = 'V': update the matrix Q of Schur vectors;
              = 'N': do not update Q.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              SELECT specifies the eigenvalues in the selected cluster. To
              select a real eigenvalue w(j), SELECT(j) must be set to
              .TRUE.. To select a complex conjugate pair of eigenvalues
              w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
              either SELECT(j) or SELECT(j+1) or both must be set to
              .TRUE.; a complex conjugate pair of eigenvalues must be
              either both included in the cluster or both excluded.

    N

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              On entry, the upper quasi-triangular matrix T, in Schur
              canonical form.
              On exit, T is overwritten by the reordered matrix T, again in
              Schur canonical form, with the selected eigenvalues in the
              leading diagonal blocks.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
              On exit, if COMPQ = 'V', Q has been postmultiplied by the
              orthogonal transformation matrix which reorders T; the
              leading M columns of Q form an orthonormal basis for the
              specified invariant subspace.
              If COMPQ = 'N', Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

    WR

              WR is DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION array, dimension (N)
              The real and imaginary parts, respectively, of the reordered
              eigenvalues of T. The eigenvalues are stored in the same
              order as on the diagonal of T, with WR(i) = T(i,i) and, if
              T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
              WI(i+1) = -WI(i). Note that if a complex eigenvalue is
              sufficiently ill-conditioned, then its value may differ
              significantly from its value before reordering.

    M

              M is INTEGER
              The dimension of the specified invariant subspace.
              0 < = M <= N.

    S

              S is DOUBLE PRECISION
              If JOB = 'E' or 'B', S is a lower bound on the reciprocal
              condition number for the selected cluster of eigenvalues.
              S cannot underestimate the true reciprocal condition number
              by more than a factor of sqrt(N). If M = 0 or N, S = 1.
              If JOB = 'N' or 'V', S is not referenced.

    SEP

              SEP is DOUBLE PRECISION
              If JOB = 'V' or 'B', SEP is the estimated reciprocal
              condition number of the specified invariant subspace. If
              M = 0 or N, SEP = norm(T).
              If JOB = 'N' or 'E', SEP is not referenced.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If JOB = 'N', LWORK >= max(1,N);
              if JOB = 'E', LWORK >= max(1,M*(N-M));
              if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If JOB = 'N' or 'E', LIWORK >= 1;
              if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              = 1: reordering of T failed because some eigenvalues are too
                   close to separate (the problem is very ill-conditioned);
                   T may have been partially reordered, and WR and WI
                   contain the eigenvalues in the same order as in T; S and
                   SEP (if requested) are set to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      DTRSEN first collects the selected eigenvalues by computing an
      orthogonal transformation Z to move them to the top left corner of T.
      In other words, the selected eigenvalues are the eigenvalues of T11
      in:
              Z**T * T * Z = ( T11 T12 ) n1
                             (  0  T22 ) n2
                                n1  n2
      where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
      of Z span the specified invariant subspace of T.
      If T has been obtained from the real Schur factorization of a matrix
      A = Q*T*Q**T, then the reordered real Schur factorization of A is given
      by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
      the corresponding invariant subspace of A.
      The reciprocal condition number of the average of the eigenvalues of
      T11 may be returned in S. S lies between 0 (very badly conditioned)
      and 1 (very well conditioned). It is computed as follows. First we
      compute R so that
                             P = ( I  R ) n1
                                 ( 0  0 ) n2
                                   n1 n2
      is the projector on the invariant subspace associated with T11.
      R is the solution of the Sylvester equation:
                            T11*R - R*T22 = T12.
      Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
      the two-norm of M. Then S is computed as the lower bound
                          (1 + F-norm(R)**2)**(-1/2)
      on the reciprocal of 2-norm(P), the true reciprocal condition number.
      S cannot underestimate 1 / 2-norm(P) by more than a factor of
      sqrt(N).
      An approximate error bound for the computed average of the
      eigenvalues of T11 is
                             EPS * norm(T) / S
      where EPS is the machine precision.
      The reciprocal condition number of the right invariant subspace
      spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
      SEP is defined as the separation of T11 and T22:
                         sep( T11, T22 ) = sigma-min( C )
      where sigma-min(C) is the smallest singular value of the
      n1*n2-by-n1*n2 matrix
         C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
      I(m) is an m by m identity matrix, and kprod denotes the Kronecker
      product. We estimate sigma-min(C) by the reciprocal of an estimate of
      the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
      cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
      When SEP is small, small changes in T can cause large changes in
      the invariant subspace. An approximate bound on the maximum angular
      error in the computed right invariant subspace is
                          EPS * norm(T) / SEP

Definition at line 311 of file dtrsen.f.

  subroutine dtrsna (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, double precision, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)

DTRSNA

Purpose:

     DTRSNA estimates reciprocal condition numbers for specified
     eigenvalues and/or right eigenvectors of a real upper
     quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q
     orthogonal).
     T must be in Schur canonical form (as returned by DHSEQR), that is,
     block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
     2-by-2 diagonal block has its diagonal elements equal and its
     off-diagonal elements of opposite sign.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies whether condition numbers are required for
              eigenvalues (S) or eigenvectors (SEP):
              = 'E': for eigenvalues only (S);
              = 'V': for eigenvectors only (SEP);
              = 'B': for both eigenvalues and eigenvectors (S and SEP).

    HOWMNY

              HOWMNY is CHARACTER*1
              = 'A': compute condition numbers for all eigenpairs;
              = 'S': compute condition numbers for selected eigenpairs
                     specified by the array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY = 'S', SELECT specifies the eigenpairs for which
              condition numbers are required. To select condition numbers
              for the eigenpair corresponding to a real eigenvalue w(j),
              SELECT(j) must be set to .TRUE.. To select condition numbers
              corresponding to a complex conjugate pair of eigenvalues w(j)
              and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
              set to .TRUE..
              If HOWMNY = 'A', SELECT is not referenced.

    N

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The upper quasi-triangular matrix T, in Schur canonical form.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    VL

              VL is DOUBLE PRECISION array, dimension (LDVL,M)
              If JOB = 'E' or 'B', VL must contain left eigenvectors of T
              (or of any Q*T*Q**T with Q orthogonal), corresponding to the
              eigenpairs specified by HOWMNY and SELECT. The eigenvectors
              must be stored in consecutive columns of VL, as returned by
              DHSEIN or DTREVC.
              If JOB = 'V', VL is not referenced.

    LDVL

              LDVL is INTEGER
              The leading dimension of the array VL.
              LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,M)
              If JOB = 'E' or 'B', VR must contain right eigenvectors of T
              (or of any Q*T*Q**T with Q orthogonal), corresponding to the
              eigenpairs specified by HOWMNY and SELECT. The eigenvectors
              must be stored in consecutive columns of VR, as returned by
              DHSEIN or DTREVC.
              If JOB = 'V', VR is not referenced.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR.
              LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.

    S

              S is DOUBLE PRECISION array, dimension (MM)
              If JOB = 'E' or 'B', the reciprocal condition numbers of the
              selected eigenvalues, stored in consecutive elements of the
              array. For a complex conjugate pair of eigenvalues two
              consecutive elements of S are set to the same value. Thus
              S(j), SEP(j), and the j-th columns of VL and VR all
              correspond to the same eigenpair (but not in general the
              j-th eigenpair, unless all eigenpairs are selected).
              If JOB = 'V', S is not referenced.

    SEP

              SEP is DOUBLE PRECISION array, dimension (MM)
              If JOB = 'V' or 'B', the estimated reciprocal condition
              numbers of the selected eigenvectors, stored in consecutive
              elements of the array. For a complex eigenvector two
              consecutive elements of SEP are set to the same value. If
              the eigenvalues cannot be reordered to compute SEP(j), SEP(j)
              is set to 0; this can only occur when the true value would be
              very small anyway.
              If JOB = 'E', SEP is not referenced.

    MM

              MM is INTEGER
              The number of elements in the arrays S (if JOB = 'E' or 'B')
               and/or SEP (if JOB = 'V' or 'B'). MM >= M.

    M

              M is INTEGER
              The number of elements of the arrays S and/or SEP actually
              used to store the estimated condition numbers.
              If HOWMNY = 'A', M is set to N.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
              If JOB = 'E', WORK is not referenced.

    LDWORK

              LDWORK is INTEGER
              The leading dimension of the array WORK.
              LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

    IWORK

              IWORK is INTEGER array, dimension (2*(N-1))
              If JOB = 'E', IWORK is not referenced.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The reciprocal of the condition number of an eigenvalue lambda is
      defined as
              S(lambda) = |v**T*u| / (norm(u)*norm(v))
      where u and v are the right and left eigenvectors of T corresponding
      to lambda; v**T denotes the transpose of v, and norm(u)
      denotes the Euclidean norm. These reciprocal condition numbers always
      lie between zero (very badly conditioned) and one (very well
      conditioned). If n = 1, S(lambda) is defined to be 1.
      An approximate error bound for a computed eigenvalue W(i) is given by
                          EPS * norm(T) / S(i)
      where EPS is the machine precision.
      The reciprocal of the condition number of the right eigenvector u
      corresponding to lambda is defined as follows. Suppose
                  T = ( lambda  c  )
                      (   0    T22 )
      Then the reciprocal condition number is
              SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
      where sigma-min denotes the smallest singular value. We approximate
      the smallest singular value by the reciprocal of an estimate of the
      one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
      defined to be abs(T(1,1)).
      An approximate error bound for a computed right eigenvector VR(i)
      is given by
                          EPS * norm(T) / SEP(i)

Definition at line 262 of file dtrsna.f.

  subroutine dtrti2 (character uplo, character diag, integer n, double precision, dimension( lda, * ) a, integer lda, integer info)

DTRTI2 computes the inverse of a triangular matrix (unblocked algorithm).

Purpose:

     DTRTI2 computes the inverse of a real upper or lower triangular
     matrix.
     This is the Level 2 BLAS version of the algorithm.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the triangular matrix A.  If UPLO = 'U', the
              leading n by n upper triangular part of the array A contains
              the upper triangular matrix, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of the array A contains
              the lower triangular matrix, and the strictly upper
              triangular part of A is not referenced.  If DIAG = 'U', the
              diagonal elements of A are also not referenced and are
              assumed to be 1.
              On exit, the (triangular) inverse of the original matrix, in
              the same storage format.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 109 of file dtrti2.f.

  subroutine dtrtri (character uplo, character diag, integer n, double precision, dimension( lda, * ) a, integer lda, integer info)

DTRTRI

Purpose:

     DTRTRI computes the inverse of a real upper or lower triangular
     matrix A.
     This is the Level 3 BLAS version of the algorithm.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the triangular matrix A.  If UPLO = 'U', the
              leading N-by-N upper triangular part of the array A contains
              the upper triangular matrix, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of the array A contains
              the lower triangular matrix, and the strictly upper
              triangular part of A is not referenced.  If DIAG = 'U', the
              diagonal elements of A are also not referenced and are
              assumed to be 1.
              On exit, the (triangular) inverse of the original matrix, in
              the same storage format.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
                   matrix is singular and its inverse can not be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 108 of file dtrtri.f.

  subroutine dtrtrs (character uplo, character trans, character diag, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DTRTRS

Purpose:

     DTRTRS solves a triangular system of the form
        A * X = B  or  A**T * X = B,
     where A is a triangular matrix of order N, and B is an N-by-NRHS
     matrix.  A check is made to verify that A is nonsingular.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular;
              = 'L':  A is lower triangular.

    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B  (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              = 'N':  A is non-unit triangular;
              = 'U':  A is unit triangular.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading N-by-N
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading N-by-N lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, if INFO = 0, the solution matrix X.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, the i-th diagonal element of A is zero,
                   indicating that the matrix is singular and the solutions
                   X have not been computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 138 of file dtrtrs.f.

  subroutine dtrttf (character transr, character uplo, integer n, double precision, dimension( 0: lda-1, 0: * ) a, integer lda, double precision, dimension( 0: * ) arf, integer info)

DTRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).

Purpose:

     DTRTTF copies a triangular matrix A from standard full format (TR)
     to rectangular full packed format (TF) .

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  ARF in Normal form is wanted;
              = 'T':  ARF in Transpose form is wanted.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

              N is INTEGER
              The order of the matrix A. N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N).
              On entry, the triangular matrix A.  If UPLO = 'U', the
              leading N-by-N upper triangular part of the array A contains
              the upper triangular matrix, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of the array A contains
              the lower triangular matrix, and the strictly upper
              triangular part of A is not referenced.

    LDA

              LDA is INTEGER
              The leading dimension of the matrix A. LDA >= max(1,N).

    ARF

              ARF is DOUBLE PRECISION array, dimension (NT).
              NT=N*(N+1)/2. On exit, the triangular matrix A in RFP format.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We first consider Rectangular Full Packed (RFP) Format when N is
      even. We give an example where N = 6.
          AP is Upper             AP is Lower
       00 01 02 03 04 05       00
          11 12 13 14 15       10 11
             22 23 24 25       20 21 22
                33 34 35       30 31 32 33
                   44 45       40 41 42 43 44
                      55       50 51 52 53 54 55
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
      three columns of AP upper. The lower triangle A(4:6,0:2) consists of
      the transpose of the first three columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:2,0:2) consists of
      the transpose of the last three columns of AP lower.
      This covers the case N even and TRANSR = 'N'.
             RFP A                   RFP A
            03 04 05                33 43 53
            13 14 15                00 44 54
            23 24 25                10 11 55
            33 34 35                20 21 22
            00 44 45                30 31 32
            01 11 55                40 41 42
            02 12 22                50 51 52
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         03 13 23 33 00 01 02    33 00 10 20 30 40 50
         04 14 24 34 44 11 12    43 44 11 21 31 41 51
         05 15 25 35 45 55 22    53 54 55 22 32 42 52
      We then consider Rectangular Full Packed (RFP) Format when N is
      odd. We give an example where N = 5.
         AP is Upper                 AP is Lower
       00 01 02 03 04              00
          11 12 13 14              10 11
             22 23 24              20 21 22
                33 34              30 31 32 33
                   44              40 41 42 43 44
      Let TRANSR = 'N'. RFP holds AP as follows:
      For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
      three columns of AP upper. The lower triangle A(3:4,0:1) consists of
      the transpose of the first two columns of AP upper.
      For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
      three columns of AP lower. The upper triangle A(0:1,1:2) consists of
      the transpose of the last two columns of AP lower.
      This covers the case N odd and TRANSR = 'N'.
             RFP A                   RFP A
            02 03 04                00 33 43
            12 13 14                10 11 44
            22 23 24                20 21 22
            00 33 34                30 31 32
            01 11 44                40 41 42
      Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
      transpose of RFP A above. One therefore gets:
               RFP A                   RFP A
         02 12 22 00 01             00 10 20 30 40 50
         03 13 23 33 11             33 11 21 31 41 51
         04 14 24 34 44             43 44 22 32 42 52

Definition at line 193 of file dtrttf.f.

  subroutine dtrttp (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) ap, integer info)

DTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).

Purpose:

     DTRTTP copies a triangular matrix A from full format (TR) to standard
     packed format (TP).

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  A is upper triangular.
              = 'L':  A is lower triangular.

    N

              N is INTEGER
              The order of the matrices AP and A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On exit, the triangular matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On exit, the upper or lower triangular matrix A, packed
              columnwise in a linear array. The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 103 of file dtrttp.f.

  subroutine dtzrzf (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DTZRZF

Purpose:

     DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
     to upper triangular form by means of orthogonal transformations.
     The upper trapezoidal matrix A is factored as
        A = ( R  0 ) * Z,
     where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
     triangular matrix.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix A.  N >= M.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading M-by-N upper trapezoidal part of the
              array A must contain the matrix to be factorized.
              On exit, the leading M-by-M upper triangular part of A
              contains the upper triangular matrix R, and elements M+1 to
              N of the first M rows of A, with the array TAU, represent the
              orthogonal matrix Z as a product of M elementary reflectors.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    TAU

              TAU is DOUBLE PRECISION array, dimension (M)
              The scalar factors of the elementary reflectors.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,M).
              For optimum performance LWORK >= M*NB, where NB is
              the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The N-by-N matrix Z can be computed by
         Z =  Z(1)*Z(2)* ... *Z(M)
      where each N-by-N Z(k) is given by
         Z(k) = I - tau(k)*v(k)*v(k)**T
      with v(k) is the kth row vector of the M-by-N matrix
         V = ( I   A(:,M+1:N) )
      I is the M-by-M identity matrix, A(:,M+1:N)
      is the output stored in A on exit from DTZRZF,
      and tau(k) is the kth element of the array TAU.

Definition at line 150 of file dtzrzf.f.

  subroutine stplqt (integer m, integer n, integer l, integer mb, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) work, integer info)

STPLQT

Purpose:

     STPLQT computes a blocked LQ factorization of a real
     'triangular-pentagonal' matrix C, which is composed of a
     triangular block A and pentagonal block B, using the compact
     WY representation for Q.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix B, and the order of the
              triangular matrix A.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size to be used in the blocked QR.  M >= MB >= 1.

    A

              A is REAL array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is REAL array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is REAL array, dimension (LDT,N)
              The lower triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.  See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

              WORK is REAL array, dimension (MB*M)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ] [ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
      upper trapezoidal matrix B2:
              [ B ] = [ B1 ] [ B2 ]
                       [ B1 ]  <- M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                [ C ] = [ A ] [ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                [ W ] = [ I ] [ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                [ V ] = [ V1 ] [ V2 ]
                       [ V1 ] <- M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The number of blocks is B = ceiling(M/MB), where each
      block is of order MB except for the last block, which is of order
      IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-N matrix T as
                   T = [T1 T2 ... TB].

Definition at line 187 of file stplqt.f.

  subroutine stplqt2 (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, integer info)

STPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

     STPLQT2 computes a LQ a factorization of a real 'triangular-pentagonal'
     matrix C, which is composed of a triangular block A and pentagonal block B,
     using the compact WY representation for Q.

Parameters
    M

              M is INTEGER
              The total number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B, and the order of
              the triangular matrix A.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    A

              A is REAL array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is REAL array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is REAL array, dimension (LDT,M)
              The N-by-N upper triangular factor T of the block reflector.
              See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= max(1,M)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ][ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
      upper trapezoidal matrix B2:
                   B = [ B1 ][ B2 ]
                       [ B1 ]  <-     M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                   C = [ A ][ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                   W = [ I ][ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                   W = [ V1 ][ V2 ]
                       [ V1 ] <-     M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The (M+N)-by-(M+N) block reflector H is then given by
                   H = I - W**T * T * W
      where W^H is the conjugate transpose of W and T is the upper triangular
      factor of the block reflector.

Definition at line 176 of file stplqt2.f.

  subroutine stpmlqt (character side, character trans, integer m, integer n, integer k, integer l, integer mb, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldt, * ) t, integer ldt, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, integer info)

STPMLQT

Purpose:

     STPMLQT applies a real orthogonal matrix Q obtained from a
     'triangular-pentagonal' real block reflector H to a general
     real matrix C, which consists of two blocks A and B.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**T from the Left;
              = 'R': apply Q or Q**T from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'T':  Transpose, apply Q**T.

    M

              M is INTEGER
              The number of rows of the matrix B. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.

    L

              L is INTEGER
              The order of the trapezoidal part of V.
              K >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size used for the storage of T.  K >= MB >= 1.
              This must be the same value of MB used to generate T
              in STPLQT.

    V

              V is REAL array, dimension (LDV,K)
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              STPLQT in B.  See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= K.

    T

              T is REAL array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by STPLQT, stored as a MB-by-K matrix.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    A

              A is REAL array, dimension
              (LDA,N) if SIDE = 'L' or
              (LDA,K) if SIDE = 'R'
              On entry, the K-by-N or M-by-K matrix A.
              On exit, A is overwritten by the corresponding block of
              Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,K);
              If SIDE = 'R', LDA >= max(1,M).

    B

              B is REAL array, dimension (LDB,N)
              On entry, the M-by-N matrix B.
              On exit, B is overwritten by the corresponding block of
              Q*C or Q**T*C or C*Q or C*Q**T.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.
              LDB >= max(1,M).

    WORK

              WORK is REAL array. The dimension of WORK is
               N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The columns of the pentagonal matrix V contain the elementary reflectors
      H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
      trapezoidal block V2:
            V = [V1] [V2].
      The size of the trapezoidal block V2 is determined by the parameter L,
      where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L
      rows of a K-by-K upper triangular matrix.  If L=K, V2 is lower triangular;
      if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
      If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                          [B]
      If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.
      The real orthogonal matrix Q is formed from V and T.
      If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
      If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
      If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
      If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.

Definition at line 212 of file stpmlqt.f.

  subroutine ztplqt (integer m, integer n, integer l, integer mb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) work, integer info)

ZTPLQT

Purpose:

     ZTPLQT computes a blocked LQ factorization of a complex
     'triangular-pentagonal' matrix C, which is composed of a
     triangular block A and pentagonal block B, using the compact
     WY representation for Q.

Parameters
    M

              M is INTEGER
              The number of rows of the matrix B, and the order of the
              triangular matrix A.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size to be used in the blocked QR.  M >= MB >= 1.

    A

              A is COMPLEX*16 array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is COMPLEX*16 array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is COMPLEX*16 array, dimension (LDT,N)
              The lower triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.  See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

              WORK is COMPLEX*16 array, dimension (MB*M)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ] [ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
      upper trapezoidal matrix B2:
              [ B ] = [ B1 ] [ B2 ]
                       [ B1 ]  <- M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                [ C ] = [ A ] [ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                [ W ] = [ I ] [ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                [ V ] = [ V1 ] [ V2 ]
                       [ V1 ] <- M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The number of blocks is B = ceiling(M/MB), where each
      block is of order MB except for the last block, which is of order
      IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-N matrix T as
                   T = [T1 T2 ... TB].

Definition at line 187 of file ztplqt.f.

  subroutine ztplqt2 (integer m, integer n, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, integer info)

ZTPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

     ZTPLQT2 computes a LQ a factorization of a complex 'triangular-pentagonal'
     matrix C, which is composed of a triangular block A and pentagonal block B,
     using the compact WY representation for Q.

Parameters
    M

              M is INTEGER
              The total number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B, and the order of
              the triangular matrix A.
              N >= 0.

    L

              L is INTEGER
              The number of rows of the lower trapezoidal part of B.
              MIN(M,N) >= L >= 0.  See Further Details.

    A

              A is COMPLEX*16 array, dimension (LDA,M)
              On entry, the lower triangular M-by-M matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the lower triangular matrix L.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

    B

              B is COMPLEX*16 array, dimension (LDB,N)
              On entry, the pentagonal M-by-N matrix B.  The first N-L columns
              are rectangular, and the last L columns are lower trapezoidal.
              On exit, B contains the pentagonal matrix V.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.  LDB >= max(1,M).

    T

              T is COMPLEX*16 array, dimension (LDT,M)
              The N-by-N upper triangular factor T of the block reflector.
              See Further Details.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= max(1,M)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The input matrix C is a M-by-(M+N) matrix
                   C = [ A ][ B ]
      where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
      matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
      upper trapezoidal matrix B2:
                   B = [ B1 ][ B2 ]
                       [ B1 ]  <-     M-by-(N-L) rectangular
                       [ B2 ]  <-     M-by-L lower trapezoidal.
      The lower trapezoidal matrix B2 consists of the first L columns of a
      N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
      B is rectangular M-by-N; if M=L=N, B is lower triangular.
      The matrix W stores the elementary reflectors H(i) in the i-th row
      above the diagonal (of A) in the M-by-(M+N) input matrix C
                   C = [ A ][ B ]
                       [ A ]  <- lower triangular M-by-M
                       [ B ]  <- M-by-N pentagonal
      so that W can be represented as
                   W = [ I ][ V ]
                       [ I ]  <- identity, M-by-M
                       [ V ]  <- M-by-N, same form as B.
      Thus, all of information needed for W is contained on exit in B, which
      we call V above.  Note that V has the same form as B; that is,
                   W = [ V1 ][ V2 ]
                       [ V1 ] <-     M-by-(N-L) rectangular
                       [ V2 ] <-     M-by-L lower trapezoidal.
      The rows of V represent the vectors which define the H(i)'s.
      The (M+N)-by-(M+N) block reflector H is then given by
                   H = I - W**T * T * W
      where W^H is the conjugate transpose of W and T is the upper triangular
      factor of the block reflector.

Definition at line 176 of file ztplqt2.f.

  subroutine ztpmlqt (character side, character trans, integer m, integer n, integer k, integer l, integer mb, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer info)

ZTPMLQT

Purpose:

     ZTPMLQT applies a complex unitary matrix Q obtained from a
     'triangular-pentagonal' complex block reflector H to a general
     complex matrix C, which consists of two blocks A and B.

Parameters
    SIDE

              SIDE is CHARACTER*1
              = 'L': apply Q or Q**H from the Left;
              = 'R': apply Q or Q**H from the Right.

    TRANS

              TRANS is CHARACTER*1
              = 'N':  No transpose, apply Q;
              = 'C':  Conjugate transpose, apply Q**H.

    M

              M is INTEGER
              The number of rows of the matrix B. M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix B. N >= 0.

    K

              K is INTEGER
              The number of elementary reflectors whose product defines
              the matrix Q.

    L

              L is INTEGER
              The order of the trapezoidal part of V.
              K >= L >= 0.  See Further Details.

    MB

              MB is INTEGER
              The block size used for the storage of T.  K >= MB >= 1.
              This must be the same value of MB used to generate T
              in ZTPLQT.

    V

              V is COMPLEX*16 array, dimension (LDV,K)
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              ZTPLQT in B.  See Further Details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= K.

    T

              T is COMPLEX*16 array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by ZTPLQT, stored as a MB-by-K matrix.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    A

              A is COMPLEX*16 array, dimension
              (LDA,N) if SIDE = 'L' or
              (LDA,K) if SIDE = 'R'
              On entry, the K-by-N or M-by-K matrix A.
              On exit, A is overwritten by the corresponding block of
              Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.
              If SIDE = 'L', LDA >= max(1,K);
              If SIDE = 'R', LDA >= max(1,M).

    B

              B is COMPLEX*16 array, dimension (LDB,N)
              On entry, the M-by-N matrix B.
              On exit, B is overwritten by the corresponding block of
              Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B.
              LDB >= max(1,M).

    WORK

              WORK is COMPLEX*16 array. The dimension of WORK is
               N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The columns of the pentagonal matrix V contain the elementary reflectors
      H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
      trapezoidal block V2:
            V = [V1] [V2].
      The size of the trapezoidal block V2 is determined by the parameter L,
      where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L
      rows of a K-by-K upper triangular matrix.  If L=K, V2 is lower triangular;
      if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
      If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                          [B]
      If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.
      The complex unitary matrix Q is formed from V and T.
      If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
      If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C.
      If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
      If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

Definition at line 212 of file ztpmlqt.f.

Author

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