realOTHERcomputational(3)

NAME
realOTHERcomputational - real

SYNOPSIS

  Functions
subroutine sbbcsd (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)
SBBCSD
subroutine sgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
SGGHD3
subroutine sgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
SGGHRD
subroutine sggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGQRF
subroutine sggrqf (m, p, n, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGRQF
subroutine sggsvp3 (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)
SGGSVP3
subroutine sgsvj0 (jobv, m, n, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info)
SGSVJ0 pre-processor for the routine sgesvj.
subroutine sgsvj1 (jobv, m, n, n1, a, lda, d, sva, mv, v, ldv, eps, sfmin, tol, nsweep, work, lwork, info)
SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only particular pivots.
subroutine shsein (side, eigsrc, initv, select, n, h, ldh, wr, wi, vl, ldvl, vr, ldvr, mm, m, work, ifaill, ifailr, info)
SHSEIN
subroutine shseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
SHSEQR
subroutine sla_lin_berr (n, nz, nrhs, res, ayb, berr)
SLA_LIN_BERR computes a component-wise relative backward error.
subroutine sla_wwaddw (n, x, y, w)
SLA_WWADDW adds a vector into a doubled-single vector.
subroutine slals0 (icompq, nl, nr, sqre, nrhs, b, ldb, bx, ldbx, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, info)
SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
subroutine slalsa (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)
SLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
subroutine slalsd (uplo, smlsiz, n, nrhs, d, e, b, ldb, rcond, rank, work, iwork, info)
SLALSD uses the singular value decomposition of A to solve the least squares problem.
real function slansf (norm, transr, uplo, n, a, work)
SLANSF
subroutine slarscl2 (m, n, d, x, ldx)
SLARSCL2 performs reciprocal diagonal scaling on a matrix.
subroutine slarz (side, m, n, l, v, incv, tau, c, ldc, work)
SLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
subroutine slarzb (side, trans, direct, storev, m, n, k, l, v, ldv, t, ldt, c, ldc, work, ldwork)
SLARZB applies a block reflector or its transpose to a general matrix.
subroutine slarzt (direct, storev, n, k, v, ldv, tau, t, ldt)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine slascl2 (m, n, d, x, ldx)
SLASCL2 performs diagonal scaling on a matrix.
subroutine slatrz (m, n, l, a, lda, tau, work)
SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine sopgtr (uplo, n, ap, tau, q, ldq, work, info)
SOPGTR
subroutine sopmtr (side, uplo, trans, m, n, ap, tau, c, ldc, work, info)
SOPMTR
subroutine sorbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)
SORBDB
subroutine sorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
SORBDB1
subroutine sorbdb2 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
SORBDB2
subroutine sorbdb3 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info)
SORBDB3
subroutine sorbdb4 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
SORBDB4
subroutine sorbdb5 (m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
SORBDB5
subroutine sorbdb6 (m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
SORBDB6
recursive subroutine sorcsd (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)
SORCSD
subroutine sorcsd2by1 (jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, iwork, info)
SORCSD2BY1
subroutine sorg2l (m, n, k, a, lda, tau, work, info)
SORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine sorg2r (m, n, k, a, lda, tau, work, info)
SORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine sorghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
SORGHR
subroutine sorgl2 (m, n, k, a, lda, tau, work, info)
SORGL2
subroutine sorglq (m, n, k, a, lda, tau, work, lwork, info)
SORGLQ
subroutine sorgql (m, n, k, a, lda, tau, work, lwork, info)
SORGQL
subroutine sorgqr (m, n, k, a, lda, tau, work, lwork, info)
SORGQR
subroutine sorgr2 (m, n, k, a, lda, tau, work, info)
SORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).
subroutine sorgrq (m, n, k, a, lda, tau, work, lwork, info)
SORGRQ
subroutine sorgtr (uplo, n, a, lda, tau, work, lwork, info)
SORGTR
subroutine sorm2l (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
SORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).
subroutine sorm2r (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).
subroutine sormbr (vect, side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
SORMBR
subroutine sormhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
SORMHR
subroutine sorml2 (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
SORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).
subroutine sormlq (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
SORMLQ
subroutine sormql (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
SORMQL
subroutine sormqr (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
SORMQR
subroutine sormr2 (side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).
subroutine sormr3 (side, trans, m, n, k, l, a, lda, tau, c, ldc, work, info)
SORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).
subroutine sormrq (side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
SORMRQ
subroutine sormrz (side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info)
SORMRZ
subroutine sormtr (side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info)
SORMTR
subroutine spbcon (uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info)
SPBCON
subroutine spbequ (uplo, n, kd, ab, ldab, s, scond, amax, info)
SPBEQU
subroutine spbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SPBRFS
subroutine spbstf (uplo, n, kd, ab, ldab, info)
SPBSTF
subroutine spbtf2 (uplo, n, kd, ab, ldab, info)
SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
subroutine spbtrf (uplo, n, kd, ab, ldab, info)
SPBTRF
subroutine spbtrs (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
SPBTRS
subroutine spftrf (transr, uplo, n, a, info)
SPFTRF
subroutine spftri (transr, uplo, n, a, info)
SPFTRI
subroutine spftrs (transr, uplo, n, nrhs, a, b, ldb, info)
SPFTRS
subroutine sppcon (uplo, n, ap, anorm, rcond, work, iwork, info)
SPPCON
subroutine sppequ (uplo, n, ap, s, scond, amax, info)
SPPEQU
subroutine spprfs (uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SPPRFS
subroutine spptrf (uplo, n, ap, info)
SPPTRF
subroutine spptri (uplo, n, ap, info)
SPPTRI
subroutine spptrs (uplo, n, nrhs, ap, b, ldb, info)
SPPTRS
subroutine spstf2 (uplo, n, a, lda, piv, rank, tol, work, info)
SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine spstrf (uplo, n, a, lda, piv, rank, tol, work, info)
SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
subroutine ssbgst (vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, info)
SSBGST
subroutine ssbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
SSBTRD
subroutine ssfrk (transr, uplo, trans, n, k, alpha, a, lda, beta, c)
SSFRK performs a symmetric rank-k operation for matrix in RFP format.
subroutine sspcon (uplo, n, ap, ipiv, anorm, rcond, work, iwork, info)
SSPCON
subroutine sspgst (itype, uplo, n, ap, bp, info)
SSPGST
subroutine ssprfs (uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SSPRFS
subroutine ssptrd (uplo, n, ap, d, e, tau, info)
SSPTRD
subroutine ssptrf (uplo, n, ap, ipiv, info)
SSPTRF
subroutine ssptri (uplo, n, ap, ipiv, work, info)
SSPTRI
subroutine ssptrs (uplo, n, nrhs, ap, ipiv, b, ldb, info)
SSPTRS
subroutine sstegr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
SSTEGR
subroutine sstein (n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifail, info)
SSTEIN
subroutine sstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
SSTEMR
subroutine stbcon (norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info)
STBCON
subroutine stbrfs (uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, ferr, berr, work, iwork, info)
STBRFS
subroutine stbtrs (uplo, trans, diag, n, kd, nrhs, ab, ldab, b, ldb, info)
STBTRS
subroutine stfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine stftri (transr, uplo, diag, n, a, info)
STFTRI
subroutine stfttp (transr, uplo, n, arf, ap, info)
STFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
subroutine stfttr (transr, uplo, n, arf, a, lda, info)
STFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
subroutine stgsen (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)
STGSEN
subroutine stgsja (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)
STGSJA
subroutine stgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
STGSNA
subroutine stpcon (norm, uplo, diag, n, ap, rcond, work, iwork, info)
STPCON
subroutine stpmqrt (side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info)
STPMQRT
subroutine stpqrt (m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)
STPQRT
subroutine stpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
STPQRT2 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 stprfs (uplo, trans, diag, n, nrhs, ap, b, ldb, x, ldx, ferr, berr, work, iwork, info)
STPRFS
subroutine stptri (uplo, diag, n, ap, info)
STPTRI
subroutine stptrs (uplo, trans, diag, n, nrhs, ap, b, ldb, info)
STPTRS
subroutine stpttf (transr, uplo, n, ap, arf, info)
STPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
subroutine stpttr (uplo, n, ap, a, lda, info)
STPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
subroutine strcon (norm, uplo, diag, n, a, lda, rcond, work, iwork, info)
STRCON
subroutine strevc (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, info)
STREVC
subroutine strevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)
STREVC3
subroutine strexc (compq, n, t, ldt, q, ldq, ifst, ilst, work, info)
STREXC
subroutine strrfs (uplo, trans, diag, n, nrhs, a, lda, b, ldb, x, ldx, ferr, berr, work, iwork, info)
STRRFS
subroutine strsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)
STRSEN
subroutine strsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)
STRSNA
subroutine strti2 (uplo, diag, n, a, lda, info)
STRTI2 computes the inverse of a triangular matrix (unblocked algorithm).
subroutine strtri (uplo, diag, n, a, lda, info)
STRTRI
subroutine strtrs (uplo, trans, diag, n, nrhs, a, lda, b, ldb, info)
STRTRS
subroutine strttf (transr, uplo, n, a, lda, arf, info)
STRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
subroutine strttp (uplo, n, a, lda, ap, info)
STRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
subroutine stzrzf (m, n, a, lda, tau, work, lwork, info)
STZRZF

Detailed Description

This is the group of real other Computational routines

Function Documentation

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

SBBCSD

Purpose:

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

    B11E

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

    B12D

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

    B12E

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

    B21D

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

    B21E

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

    B22D

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

    B22E

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

    WORK

              WORK is REAL 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 SBBCSD did not converge, INFO specifies the number
                    of nonzero entries in PHI, and B11D, B11E, etc.,
                    contain the partially reduced matrix.

Internal Parameters:

      TOLMUL  REAL, 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 sbbcsd.f.

  subroutine sgghd3 (character compq, character compz, integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

SGGHD3

Purpose:

     SGGHD3 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 SGGHD3 reduces the original
     problem to generalized Hessenberg form.
     This is a blocked variant of SGGHRD, 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 SGGBAL; 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 sgghd3.f.

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

SGGHRD

Purpose:

     SGGHRD 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 SGGHRD 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 SGGBAL; 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 REAL 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 REAL 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 REAL 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 REAL 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 sgghrd.f.

  subroutine sggqrf (integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( * ) taua, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) taub, real, dimension( * ) work, integer lwork, integer info)

SGGQRF

Purpose:

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

    B

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

    WORK

              WORK is REAL 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 SORMQR.
              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 SORGQR.
      To use Q to update another matrix, use LAPACK subroutine SORMQR.
      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 SORGRQ.
      To use Z to update another matrix, use LAPACK subroutine SORMRQ.

Definition at line 213 of file sggqrf.f.

  subroutine sggrqf (integer m, integer p, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) taua, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) taub, real, dimension( * ) work, integer lwork, integer info)

SGGRQF

Purpose:

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

    B

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

    WORK

              WORK is REAL 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 SORMRQ.
              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 SORGRQ.
      To use Q to update another matrix, use LAPACK subroutine SORMRQ.
      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 SORGQR.
      To use Z to update another matrix, use LAPACK subroutine SORMQR.

Definition at line 212 of file sggrqf.f.

  subroutine sggsvp3 (character jobu, character jobv, character jobq, integer m, integer p, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real tola, real tolb, integer k, integer l, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SGGSVP3

Purpose:

     SGGSVP3 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
     SGGSVD3.

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 REAL 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 REAL 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 REAL

    TOLB

              TOLB is REAL
              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 REAL 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 REAL 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 REAL 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 REAL array, dimension (N)

    WORK

              WORK is REAL 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 SGEQP3 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.
      SGGSVP3 replaces the deprecated subroutine SGGSVP.

Definition at line 269 of file sggsvp3.f.

  subroutine sgsvj0 (character*1 jobv, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( n ) d, real, dimension( n ) sva, integer mv, real, dimension( ldv, * ) v, integer ldv, real eps, real sfmin, real tol, integer nsweep, real, dimension( lwork ) work, integer lwork, integer info)

SGSVJ0 pre-processor for the routine sgesvj.

Purpose:

     SGSVJ0 is called from SGESVJ as a pre-processor and that is its main
     purpose. It applies Jacobi rotations in the same way as SGESVJ 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
              EPS = SLAMCH('Epsilon')

    SFMIN

              SFMIN is REAL
              SFMIN = SLAMCH('Safe Minimum')

    TOL

              TOL is REAL
              TOL is the threshold for Jacobi rotations. For a pair
              A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
              applied only if ABS(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 REAL 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:
    SGSVJ0 is used just to enable SGESVJ 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 sgsvj0.f.

  subroutine sgsvj1 (character*1 jobv, integer m, integer n, integer n1, real, dimension( lda, * ) a, integer lda, real, dimension( n ) d, real, dimension( n ) sva, integer mv, real, dimension( ldv, * ) v, integer ldv, real eps, real sfmin, real tol, integer nsweep, real, dimension( lwork ) work, integer lwork, integer info)

SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only particular pivots.

Purpose:

     SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
     purpose. It applies Jacobi rotations in the same way as SGESVJ 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
     ~~~~~~~~~~~~~~~
     SGSVJ1 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
              EPS = SLAMCH('Epsilon')

    SFMIN

              SFMIN is REAL
              SFMIN = SLAMCH('Safe Minimum')

    TOL

              TOL is REAL
              TOL is the threshold for Jacobi rotations. For a pair
              A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
              applied only if ABS(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 REAL 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 sgsvj1.f.

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

SHSEIN

Purpose:

     SHSEIN 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 SHSEQR; 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 SHSEIN 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, SHSEIN 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 REAL 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 REAL array, dimension (N)

    WI

              WI is REAL 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 REAL 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 REAL 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 REAL 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 shsein.f.

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

SHSEQR

Purpose:

        SHSEQR 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 SGEBAL, and then passed to ZGEHRD
               when the matrix output by SGEBAL 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 REAL 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 SHSEQR, 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 REAL array, dimension (N)

    WI

              WI is REAL 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 REAL 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 SORGHR
               after the call to SGEHRD 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 REAL 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 SHSEQR does a workspace query.
               In this case, SHSEQR 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, SHSEQR 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,'SHSEQR',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 SLAHQR vs SLAQR0 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
                           SLAHQR 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 shseqr.f.

  subroutine sla_lin_berr (integer n, integer nz, integer nrhs, real, dimension( n, nrhs ) res, real, dimension( n, nrhs ) ayb, real, dimension( nrhs ) berr)

SLA_LIN_BERR computes a component-wise relative backward error.

Purpose:

        SLA_LIN_BERR computes componentwise 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 componentwise 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 REAL array, dimension (N,NRHS)
         The residual matrix, i.e., the matrix R in the relative backward
         error formula above.

    AYB

              AYB is REAL 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 sla_gerfsx_extended.f).

    BERR

              BERR is REAL array, dimension (NRHS)
         The componentwise 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 sla_lin_berr.f.

  subroutine sla_wwaddw (integer n, real, dimension( * ) x, real, dimension( * ) y, real, dimension( * ) w)

SLA_WWADDW adds a vector into a doubled-single vector.

Purpose:

        SLA_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 REAL array, dimension (N)
                The first part of the doubled-single accumulation vector.

    Y

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

    W

              W is REAL 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 sla_wwaddw.f.

  subroutine slals0 (integer icompq, integer nl, integer nr, integer sqre, integer nrhs, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldbx, * ) bx, integer ldbx, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, real, dimension( ldgnum, * ) givnum, integer ldgnum, real, dimension( ldgnum, * ) poles, real, dimension( * ) difl, real, dimension( ldgnum, * ) difr, real, dimension( * ) z, integer k, real c, real s, real, dimension( * ) work, integer info)

SLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:

     SLALS0 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL
             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 REAL
             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 REAL 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 slals0.f.

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

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

Purpose:

     SLALSA 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, SLALSA applies the inverse of the left singular vector
     matrix of an upper bidiagonal matrix to the right hand side; and if
     ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
     right hand side. The singular vector matrices were generated in
     compact form by SLALSA.

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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension ( LDU, NLVL ).
             where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

    DIFR

              DIFR is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 slalsa.f.

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

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

Purpose:

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

    B

              B is REAL 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 REAL
             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 REAL 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 slalsd.f.

  real function slansf (character norm, character transr, character uplo, integer n, real, dimension( 0: * ) a, real, dimension( 0: * ) work)

SLANSF

Purpose:

     SLANSF 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
    SLANSF

        SLANSF = ( 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 SLANSF 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, SLANSF is
              set to zero.

    A

              A is REAL 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 REAL 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 slansf.f.

  subroutine slarscl2 (integer m, integer n, real, dimension( * ) d, real, dimension( ldx, * ) x, integer ldx)

SLARSCL2 performs reciprocal diagonal scaling on a matrix.

Purpose:

     SLARSCL2 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_sge_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 REAL array, length M
         Diagonal matrix D, stored as a vector of length M.

    X

              X is REAL 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 slarscl2.f.

  subroutine slarz (character side, integer m, integer n, integer l, real, dimension( * ) v, integer incv, real tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work)

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

Purpose:

     SLARZ 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 STZRZF.

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

    INCV

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

    TAU

              TAU is REAL
              The value tau in the representation of H.

    C

              C is REAL 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 REAL 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 slarz.f.

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

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

Purpose:

     SLARZB 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 REAL 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 REAL 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 REAL 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 REAL 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 slarzb.f.

  subroutine slarzt (character direct, character storev, integer n, integer k, real, dimension( ldv, * ) v, integer ldv, real, dimension( * ) tau, real, dimension( ldt, * ) t, integer ldt)

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

Purpose:

     SLARZT 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 REAL 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is REAL 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 slarzt.f.

  subroutine slascl2 (integer m, integer n, real, dimension( * ) d, real, dimension( ldx, * ) x, integer ldx)

SLASCL2 performs diagonal scaling on a matrix.

Purpose:

     SLASCL2 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_sge_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 REAL array, length M
         Diagonal matrix D, stored as a vector of length M.

    X

              X is REAL 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 slascl2.f.

  subroutine slatrz (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work)

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

Purpose:

     SLATRZ 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 REAL 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 REAL array, dimension (M)
              The scalar factors of the elementary reflectors.

    WORK

              WORK is REAL 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 slatrz.f.

  subroutine sopgtr (character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) tau, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer info)

SOPGTR

Purpose:

     SOPGTR 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
     SSPTRD 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 SSPTRD;
              = 'L': Lower triangular packed storage used in previous
                     call to SSPTRD.

    N

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

    AP

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

    TAU

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

    Q

              Q is REAL 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 REAL 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 sopgtr.f.

  subroutine sopmtr (character side, character uplo, character trans, integer m, integer n, real, dimension( * ) ap, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SOPMTR

Purpose:

     SOPMTR 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 SSPTRD 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 SSPTRD;
              = 'L': Lower triangular packed storage used in previous
                     call to SSPTRD.

    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 REAL 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 SSPTRD.  AP is modified by the routine but
              restored on exit.

    TAU

              TAU is REAL 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 SSPTRD.

    C

              C is REAL 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 REAL 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 sopmtr.f.

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

SORBDB

Purpose:

     SORBDB 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 SORCSD
     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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (P)
              The scalar factors of the elementary reflectors that define
              P1.

    TAUP2

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

    TAUQ1

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

    TAUQ2

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

    WORK

              WORK is REAL 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 SORCSD for details.
      P1, P2, Q1, and Q2 are represented as products of elementary
      reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2
      using SORGQR and SORGLQ.

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

Definition at line 284 of file sorbdb.f.

  subroutine sorbdb1 (integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real, dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1, real, dimension(*) work, integer lwork, integer info)

SORBDB1

Purpose:

     SORBDB1 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 SORBDB2, SORBDB3, and SORBDB4 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 REAL 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 REAL 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 REAL array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

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

    TAUP1

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

    TAUP2

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

    TAUQ1

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

    WORK

              WORK is REAL 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 SORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
      and SORGLQ.

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

Definition at line 201 of file sorbdb1.f.

  subroutine sorbdb2 (integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real, dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1, real, dimension(*) work, integer lwork, integer info)

SORBDB2

Purpose:

     SORBDB2 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 SORBDB1, SORBDB3, and SORBDB4 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 REAL 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 REAL 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 REAL array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

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

    TAUP1

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

    TAUP2

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

    TAUQ1

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

    WORK

              WORK is REAL 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 SORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
      and SORGLQ.

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

Definition at line 199 of file sorbdb2.f.

  subroutine sorbdb3 (integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real, dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1, real, dimension(*) work, integer lwork, integer info)

SORBDB3

Purpose:

     SORBDB3 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 SORBDB1, SORBDB2, and SORBDB4 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 REAL 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 REAL 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 REAL array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

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

    TAUP1

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

    TAUP2

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

    TAUQ1

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

    WORK

              WORK is REAL 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 SORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
      and SORGLQ.

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

Definition at line 200 of file sorbdb3.f.

  subroutine sorbdb4 (integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real, dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1, real, dimension(*) phantom, real, dimension(*) work, integer lwork, integer info)

SORBDB4

Purpose:

     SORBDB4 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 SORBDB1, SORBDB2, and SORBDB3 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 REAL 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 REAL 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 REAL array, dimension (Q)
               The entries of the bidiagonal blocks B11, B21 are defined by
               THETA and PHI. See Further Details.

    PHI

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

    TAUP1

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

    TAUP2

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

    TAUQ1

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

    PHANTOM

              PHANTOM is REAL 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 REAL 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 SORCSD for details.
      P1, P2, and Q1 are represented as products of elementary reflectors.
      See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
      and SORGLQ.

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

Definition at line 211 of file sorbdb4.f.

  subroutine sorbdb5 (integer m1, integer m2, integer n, real, dimension(*) x1, integer incx1, real, dimension(*) x2, integer incx2, real, dimension(ldq1,*) q1, integer ldq1, real, dimension(ldq2,*) q2, integer ldq2, real, dimension(*) work, integer lwork, integer info)

SORBDB5

Purpose:

     SORBDB5 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 sorbdb5.f.

  subroutine sorbdb6 (integer m1, integer m2, integer n, real, dimension(*) x1, integer incx1, real, dimension(*) x2, integer incx2, real, dimension(ldq1,*) q1, integer ldq1, real, dimension(ldq2,*) q2, integer ldq2, real, dimension(*) work, integer lwork, integer info)

SORBDB6

Purpose:

     SORBDB6 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 sorbdb6.f.

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

SORCSD

Purpose:

     SORCSD 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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:  SBBCSD 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 sorcsd.f.

  subroutine sorcsd2by1 (character jobu1, character jobu2, character jobv1t, integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(ldu1,*) u1, integer ldu1, real, dimension(ldu2,*) u2, integer ldu2, real, dimension(ldv1t,*) v1t, integer ldv1t, real, dimension(*) work, integer lwork, integer, dimension(*) iwork, integer info)

SORCSD2BY1

Purpose:

     SORCSD2BY1 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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:  SBBCSD 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 sorcsd2by1.f.

  subroutine sorg2l (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)

SORG2L generates all or part of the orthogonal matrix Q from a QL factorization determined by sgeqlf (unblocked algorithm).

Purpose:

     SORG2L 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 SGEQLF.

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 REAL 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 SGEQLF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQLF.

    WORK

              WORK is REAL 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 sorg2l.f.

  subroutine sorg2r (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)

SORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf (unblocked algorithm).

Purpose:

     SORG2R 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 SGEQRF.

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 REAL 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 SGEQRF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQRF.

    WORK

              WORK is REAL 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 sorg2r.f.

  subroutine sorghr (integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SORGHR

Purpose:

     SORGHR generates a real orthogonal matrix Q which is defined as the
     product of IHI-ILO elementary reflectors of order N, as returned by
     SGEHRD:
     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 SGEHRD. 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 REAL array, dimension (LDA,N)
              On entry, the vectors which define the elementary reflectors,
              as returned by SGEHRD.
              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 REAL array, dimension (N-1)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEHRD.

    WORK

              WORK is REAL 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 sorghr.f.

  subroutine sorgl2 (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)

SORGL2

Purpose:

     SORGL2 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 SGELQF.

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 REAL 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 SGELQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGELQF.

    WORK

              WORK is REAL 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 sorgl2.f.

  subroutine sorglq (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SORGLQ

Purpose:

     SORGLQ 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 SGELQF.

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 REAL 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 SGELQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGELQF.

    WORK

              WORK is REAL 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 sorglq.f.

  subroutine sorgql (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SORGQL

Purpose:

     SORGQL 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 SGEQLF.

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 REAL 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 SGEQLF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQLF.

    WORK

              WORK is REAL 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 sorgql.f.

  subroutine sorgqr (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SORGQR

Purpose:

     SORGQR 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 SGEQRF.

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 REAL 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 SGEQRF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQRF.

    WORK

              WORK is REAL 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 sorgqr.f.

  subroutine sorgr2 (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)

SORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf (unblocked algorithm).

Purpose:

     SORGR2 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 SGERQF.

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 REAL 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 SGERQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGERQF.

    WORK

              WORK is REAL 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 sorgr2.f.

  subroutine sorgrq (integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SORGRQ

Purpose:

     SORGRQ 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 SGERQF.

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 REAL 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 SGERQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGERQF.

    WORK

              WORK is REAL 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 sorgrq.f.

  subroutine sorgtr (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SORGTR

Purpose:

     SORGTR generates a real orthogonal matrix Q which is defined as the
     product of n-1 elementary reflectors of order N, as returned by
     SSYTRD:
     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 SSYTRD;
              = 'L': Lower triangle of A contains elementary reflectors
                     from SSYTRD.

    N

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

    A

              A is REAL array, dimension (LDA,N)
              On entry, the vectors which define the elementary reflectors,
              as returned by SSYTRD.
              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 REAL array, dimension (N-1)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SSYTRD.

    WORK

              WORK is REAL 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 sorgtr.f.

  subroutine sorm2l (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SORM2L multiplies a general matrix by the orthogonal matrix from a QL factorization determined by sgeqlf (unblocked algorithm).

Purpose:

     SORM2L 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 SGEQLF. 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 REAL 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
              SGEQLF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQLF.

    C

              C is REAL 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 REAL 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 sorm2l.f.

  subroutine sorm2r (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sgeqrf (unblocked algorithm).

Purpose:

     SORM2R 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 SGEQRF. 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 REAL 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
              SGEQRF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQRF.

    C

              C is REAL 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 REAL 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 sorm2r.f.

  subroutine sormbr (character vect, character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMBR

Purpose:

     If VECT = 'Q', SORMBR 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', SORMBR 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 SGEBRD 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 SGEBRD.
              If VECT = 'P', the number of rows in the original
              matrix reduced by SGEBRD.
              K >= 0.

    A

              A is REAL 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 SGEBRD.

    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 REAL 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 SGEBRD in the array argument TAUQ or TAUP.

    C

              C is REAL 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 REAL 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 194 of file sormbr.f.

  subroutine sormhr (character side, character trans, integer m, integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMHR

Purpose:

     SORMHR 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 SGEHRD:
     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 SGEHRD. 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 REAL array, dimension
                                   (LDA,M) if SIDE = 'L'
                                   (LDA,N) if SIDE = 'R'
              The vectors which define the elementary reflectors, as
              returned by SGEHRD.

    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 REAL 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 SGEHRD.

    C

              C is REAL 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 REAL 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 177 of file sormhr.f.

  subroutine sorml2 (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SORML2 multiplies a general matrix by the orthogonal matrix from a LQ factorization determined by sgelqf (unblocked algorithm).

Purpose:

     SORML2 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 SGELQF. 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 REAL 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
              SGELQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGELQF.

    C

              C is REAL 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 REAL 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 sorml2.f.

  subroutine sormlq (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMLQ

Purpose:

     SORMLQ 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 SGELQF. 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 REAL 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
              SGELQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGELQF.

    C

              C is REAL 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 REAL 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 166 of file sormlq.f.

  subroutine sormql (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMQL

Purpose:

     SORMQL 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 SGEQLF. 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 REAL 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
              SGEQLF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQLF.

    C

              C is REAL 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 REAL 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 166 of file sormql.f.

  subroutine sormqr (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMQR

Purpose:

     SORMQR 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 SGEQRF. 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 REAL 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
              SGEQRF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGEQRF.

    C

              C is REAL 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 REAL 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 166 of file sormqr.f.

  subroutine sormr2 (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sgerqf (unblocked algorithm).

Purpose:

     SORMR2 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 SGERQF. 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 REAL 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
              SGERQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGERQF.

    C

              C is REAL 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 REAL 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 sormr2.f.

  subroutine sormr3 (character side, character trans, integer m, integer n, integer k, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SORMR3 multiplies a general matrix by the orthogonal matrix from a RZ factorization determined by stzrzf (unblocked algorithm).

Purpose:

     SORMR3 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 STZRZF. 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 REAL 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
              STZRZF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by STZRZF.

    C

              C is REAL 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 REAL 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 sormr3.f.

  subroutine sormrq (character side, character trans, integer m, integer n, integer k, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMRQ

Purpose:

     SORMRQ 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 SGERQF. 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 REAL 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
              SGERQF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by SGERQF.

    C

              C is REAL 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 REAL 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 166 of file sormrq.f.

  subroutine sormrz (character side, character trans, integer m, integer n, integer k, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMRZ

Purpose:

     SORMRZ 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 STZRZF. 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 REAL 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
              STZRZF 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 REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i), as returned by STZRZF.

    C

              C is REAL 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 REAL 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 sormrz.f.

  subroutine sormtr (character side, character uplo, character trans, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SORMTR

Purpose:

     SORMTR 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 SSYTRD:
     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 SSYTRD;
              = 'L': Lower triangle of A contains elementary reflectors
                     from SSYTRD.

    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 REAL array, dimension
                                   (LDA,M) if SIDE = 'L'
                                   (LDA,N) if SIDE = 'R'
              The vectors which define the elementary reflectors, as
              returned by SSYTRD.

    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 REAL 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 SSYTRD.

    C

              C is REAL 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 REAL 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 170 of file sormtr.f.

  subroutine spbcon (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real anorm, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SPBCON

Purpose:

     SPBCON 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 SPBTRF.
     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 REAL 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 REAL
              The 1-norm (or infinity-norm) of the symmetric band matrix A.

    RCOND

              RCOND is REAL
              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 REAL 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 spbcon.f.

  subroutine spbequ (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) s, real scond, real amax, integer info)

SPBEQU

Purpose:

     SPBEQU 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 REAL 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 REAL array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

    SCOND

              SCOND is REAL
              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 REAL
              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 spbequ.f.

  subroutine spbrfs (character uplo, integer n, integer kd, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldafb, * ) afb, integer ldafb, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SPBRFS

Purpose:

     SPBRFS 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 REAL 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 REAL 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
              SPBTRF, 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 REAL 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 REAL array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by SPBTRS.
              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 REAL 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 REAL 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 REAL 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 spbrfs.f.

  subroutine spbstf (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info)

SPBSTF

Purpose:

     SPBSTF computes a split Cholesky factorization of a real
     symmetric positive definite band matrix A.
     This routine is designed to be used in conjunction with SSBGST.
     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 REAL 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 spbstf.f.

  subroutine spbtf2 (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info)

SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

Purpose:

     SPBTF2 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 REAL 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 spbtf2.f.

  subroutine spbtrf (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info)

SPBTRF

Purpose:

     SPBTRF 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 REAL 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 spbtrf.f.

  subroutine spbtrs (character uplo, integer n, integer kd, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldb, * ) b, integer ldb, integer info)

SPBTRS

Purpose:

     SPBTRS 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 SPBTRF.

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 REAL 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 REAL 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 spbtrs.f.

  subroutine spftrf (character transr, character uplo, integer n, real, dimension( 0: * ) a, integer info)

SPFTRF

Purpose:

     SPFTRF 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 REAL 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 spftrf.f.

  subroutine spftri (character transr, character uplo, integer n, real, dimension( 0: * ) a, integer info)

SPFTRI

Purpose:

     SPFTRI 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 SPFTRF.

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 REAL 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 spftri.f.

  subroutine spftrs (character transr, character uplo, integer n, integer nrhs, real, dimension( 0: * ) a, real, dimension( ldb, * ) b, integer ldb, integer info)

SPFTRS

Purpose:

     SPFTRS 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 SPFTRF.

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 REAL array, dimension ( N*(N+1)/2 )
              The triangular factor U or L from the Cholesky factorization
              of RFP A = U**H*U or RFP A = L*L**T, as computed by SPFTRF.
              See note below for more details about RFP A.

    B

              B is REAL 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 spftrs.f.

  subroutine sppcon (character uplo, integer n, real, dimension( * ) ap, real anorm, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SPPCON

Purpose:

     SPPCON 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
     SPPTRF.
     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 REAL 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 REAL
              The 1-norm (or infinity-norm) of the symmetric matrix A.

    RCOND

              RCOND is REAL
              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 REAL 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 sppcon.f.

  subroutine sppequ (character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) s, real scond, real amax, integer info)

SPPEQU

Purpose:

     SPPEQU 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 REAL 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 REAL array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

    SCOND

              SCOND is REAL
              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 REAL
              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 sppequ.f.

  subroutine spprfs (character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SPPRFS

Purpose:

     SPPRFS 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 REAL 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 REAL 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 SPPTRF/CPPTRF,
              packed columnwise in a linear array in the same format as A
              (see AP).

    B

              B is REAL 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 REAL array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by SPPTRS.
              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 REAL 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 REAL 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 REAL 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 spprfs.f.

  subroutine spptrf (character uplo, integer n, real, dimension( * ) ap, integer info)

SPPTRF

Purpose:

     SPPTRF 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 REAL 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 spptrf.f.

  subroutine spptri (character uplo, integer n, real, dimension( * ) ap, integer info)

SPPTRI

Purpose:

     SPPTRI 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 SPPTRF.

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 REAL 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 spptri.f.

  subroutine spptrs (character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( ldb, * ) b, integer ldb, integer info)

SPPTRS

Purpose:

     SPPTRS 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 SPPTRF.

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 REAL 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 REAL 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 spptrs.f.

  subroutine spstf2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( n ) piv, integer rank, real tol, real, dimension( 2*n ) work, integer info)

SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

Purpose:

     SPSTF2 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 REAL 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 REAL
              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 REAL 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 spstf2.f.

  subroutine spstrf (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( n ) piv, integer rank, real tol, real, dimension( 2*n ) work, integer info)

SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

Purpose:

     SPSTRF 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 REAL 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 REAL
              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 REAL 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 spstrf.f.

  subroutine ssbgst (character vect, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) work, integer info)

SSBGST

Purpose:

     SSBGST 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 SPBSTF, 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 REAL 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 REAL array, dimension (LDBB,N)
              The banded factor S from the split Cholesky factorization of
              B, as returned by SPBSTF, 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 REAL 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 REAL 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 ssbgst.f.

  subroutine ssbtrd (character vect, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer info)

SSBTRD

Purpose:

     SSBTRD 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 REAL 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 REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T.

    E

              E is REAL 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 REAL 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 REAL 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 ssbtrd.f.

  subroutine ssfrk (character transr, character uplo, character trans, integer n, integer k, real alpha, real, dimension( lda, * ) a, integer lda, real beta, real, dimension( * ) c)

SSFRK performs a symmetric rank-k operation for matrix in RFP format.

Purpose:

     Level 3 BLAS like routine for C in RFP Format.
     SSFRK 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 REAL
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    A

              A is REAL 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 REAL
               On entry, BETA specifies the scalar beta.
               Unchanged on exit.

    C

              C is REAL 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 ssfrk.f.

  subroutine sspcon (character uplo, integer n, real, dimension( * ) ap, integer, dimension( * ) ipiv, real anorm, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSPCON

Purpose:

     SSPCON 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 SSPTRF.
     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 REAL 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 SSPTRF, 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 SSPTRF.

    ANORM

              ANORM is REAL
              The 1-norm of the original matrix A.

    RCOND

              RCOND is REAL
              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 REAL 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 sspcon.f.

  subroutine sspgst (integer itype, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, integer info)

SSPGST

Purpose:

     SSPGST 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 SPPTRF.

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 REAL 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 REAL 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 SPPTRF.

    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 sspgst.f.

  subroutine ssprfs (character uplo, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( * ) afp, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSPRFS

Purpose:

     SSPRFS 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 REAL 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 REAL 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 SSPTRF, 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 SSPTRF.

    B

              B is REAL 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 REAL array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by SSPTRS.
              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 REAL 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 REAL 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 REAL 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 ssprfs.f.

  subroutine ssptrd (character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, integer info)

SSPTRD

Purpose:

     SSPTRD 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 REAL 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 REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is REAL 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 REAL 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 ssptrd.f.

  subroutine ssptrf (character uplo, integer n, real, dimension( * ) ap, integer, dimension( * ) ipiv, integer info)

SSPTRF

Purpose:

     SSPTRF 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 REAL 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:

      5-96 - Based on modifications by J. Lewis, Boeing Computer Services
             Company
      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).

Definition at line 156 of file ssptrf.f.

  subroutine ssptri (character uplo, integer n, real, dimension( * ) ap, integer, dimension( * ) ipiv, real, dimension( * ) work, integer info)

SSPTRI

Purpose:

     SSPTRI 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 SSPTRF.

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 REAL 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 SSPTRF,
              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 SSPTRF.

    WORK

              WORK is REAL 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 ssptri.f.

  subroutine ssptrs (character uplo, integer n, integer nrhs, real, dimension( * ) ap, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, integer info)

SSPTRS

Purpose:

     SSPTRS 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 SSPTRF.

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 REAL 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 SSPTRF, 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 SSPTRF.

    B

              B is REAL 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 ssptrs.f.

  subroutine sstegr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSTEGR

Purpose:

     SSTEGR 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.
     SSTEGR is a compatibility wrapper around the improved SSTEMR routine.
     See SSTEMR for further details.
     One important change is that the ABSTOL parameter no longer provides any
     benefit and hence is no longer used.
     Note : SSTEGR and SSTEMR 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 REAL array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is REAL 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 REAL
              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 REAL
              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 REAL
              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 REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is REAL 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 REAL 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 SLARRE,
                    if INFO = 2X, internal error in SLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by SLARRE or
                    SLARRV, 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 sstegr.f.

  subroutine sstein (integer n, real, dimension( * ) d, real, dimension( * ) e, integer m, real, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSTEIN

Purpose:

     SSTEIN 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 REAL array, dimension (N)
              The n diagonal elements of the tridiagonal matrix T.

    E

              E is REAL 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 REAL 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 SSTEBZ 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 SSTEBZ 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 SSTEBZ is expected here. )

    Z

              Z is REAL 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 REAL 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 sstein.f.

  subroutine sstemr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical tryrac, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSTEMR

Purpose:

     SSTEMR 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.SSTEMR 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 REAL array, dimension (N)
              On entry, the N diagonal elements of the tridiagonal matrix
              T. On exit, D is overwritten.

    E

              E is REAL 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 REAL
              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 REAL
              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 REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is REAL 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 REAL 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 SLARRE,
                    if INFO = 2X, internal error in SLARRV.
                    Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                    the nonzero error code returned by SLARRE or
                    SLARRV, 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 sstemr.f.

  subroutine stbcon (character norm, character uplo, character diag, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

STBCON

Purpose:

     STBCON 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 REAL 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 REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is REAL 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 stbcon.f.

  subroutine stbrfs (character uplo, character trans, character diag, integer n, integer kd, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

STBRFS

Purpose:

     STBRFS 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 STBTRS or some other
     means before entering this routine.  STBRFS 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 stbrfs.f.

  subroutine stbtrs (character uplo, character trans, character diag, integer n, integer kd, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldb, * ) b, integer ldb, integer info)

STBTRS

Purpose:

     STBTRS 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 REAL 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 REAL 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 stbtrs.f.

  subroutine stfsm (character transr, character side, character uplo, character trans, character diag, integer m, integer n, real alpha, real, dimension( 0: * ) a, real, dimension( 0: ldb-1, 0: * ) b, integer ldb)

STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

     Level 3 BLAS like routine for A in RFP Format.
     STFSM  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 REAL
               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 REAL 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 REAL 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 stfsm.f.

  subroutine stftri (character transr, character uplo, character diag, integer n, real, dimension( 0: * ) a, integer info)

STFTRI

Purpose:

     STFTRI 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 REAL array, dimension (NT);
              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 stftri.f.

  subroutine stfttp (character transr, character uplo, integer n, real, dimension( 0: * ) arf, real, dimension( 0: * ) ap, integer info)

STFTTP copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).

Purpose:

     STFTTP 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 REAL 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 REAL 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 stfttp.f.

  subroutine stfttr (character transr, character uplo, integer n, real, dimension( 0: * ) arf, real, dimension( 0: lda-1, 0: * ) a, integer lda, integer info)

STFTTR copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).

Purpose:

     STFTTR 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 REAL 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 REAL 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 stfttr.f.

  subroutine stgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, integer m, real pl, real pr, real, dimension( * ) dif, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

STGSEN

Purpose:

     STGSEN 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 SGGES), i.e. A is block upper
     triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
     triangular.
     STGSEN also computes the generalized eigenvalues
                 w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
     of the reordered matrix pair (A, B).
     Optionally, STGSEN 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 REAL 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 REAL 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 REAL array, dimension (N)

    ALPHAI

              ALPHAI is REAL array, dimension (N)

    BETA

              BETA is REAL 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 REAL 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 REAL 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 REAL

    PR

              PR is REAL
              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 REAL 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 REAL 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:

      STGSEN 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 SLATDF), then the parameter
      IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
      (IJOB = 2 will be used)). See STGSYL 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 stgsen.f.

  subroutine stgsja (character jobu, character jobv, character jobq, integer m, integer p, integer n, integer k, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real tola, real tolb, real, dimension( * ) alpha, real, dimension( * ) beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer ncycle, integer info)

STGSJA

Purpose:

     STGSJA 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 SGGSVP
     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 STGSJA.
              See Further Details.

    A

              A is REAL 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 REAL 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 REAL

    TOLB

              TOLB is REAL
              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)*MACHEPS,
                  TOLB = max(P,N)*norm(B)*MACHEPS.

    ALPHA

              ALPHA is REAL array, dimension (N)

    BETA

              BETA is REAL 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 REAL array, dimension (LDU,M)
              On entry, if JOBU = 'U', U must contain a matrix U1 (usually
              the orthogonal matrix returned by SGGSVP).
              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 REAL array, dimension (LDV,P)
              On entry, if JOBV = 'V', V must contain a matrix V1 (usually
              the orthogonal matrix returned by SGGSVP).
              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 REAL array, dimension (LDQ,N)
              On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
              the orthogonal matrix returned by SGGSVP).
              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 REAL 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:

      STGSJA 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 stgsja.f.

  subroutine stgsna (character job, character howmny, logical, dimension( * ) select, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) dif, integer mm, integer m, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

STGSNA

Purpose:

     STGSNA 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 SGGES),
     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 REAL 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 REAL 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 REAL 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 STGEVC.
              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 REAL 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 STGEVC.
              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 REAL 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 REAL 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 REAL 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 SLATDF), then the parameter DIFDRI (see below) should be
         changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
         See STGSYL 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
         SLATDF), then the parameter DIFDRI (see below) should be changed
         from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
         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 stgsna.f.

  subroutine stpcon (character norm, character uplo, character diag, integer n, real, dimension( * ) ap, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

STPCON

Purpose:

     STPCON 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 REAL 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 REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is REAL 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 stpcon.f.

  subroutine stpmqrt (character side, character trans, integer m, integer n, integer k, integer l, integer nb, 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)

STPMQRT

Purpose:

     STPMQRT 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 REAL 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 REAL 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 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', LDC >= max(1,K);
              If SIDE = 'R', LDC >= 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*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 stpmqrt.f.

  subroutine stpqrt (integer m, integer n, integer l, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) work, integer info)

STPQRT

Purpose:

     STPQRT 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 REAL 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 REAL 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 REAL 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 REAL 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 stpqrt.f.

  subroutine stpqrt2 (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)

STPQRT2 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:

     STPQRT2 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 REAL 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 REAL 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 REAL 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^H
      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 stpqrt2.f.

  subroutine stprfs (character uplo, character trans, character diag, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

STPRFS

Purpose:

     STPRFS 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 STPTRS or some other
     means before entering this routine.  STPRFS 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 stprfs.f.

  subroutine stptri (character uplo, character diag, integer n, real, dimension( * ) ap, integer info)

STPTRI

Purpose:

     STPTRI 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 REAL 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 stptri.f.

  subroutine stptrs (character uplo, character trans, character diag, integer n, integer nrhs, real, dimension( * ) ap, real, dimension( ldb, * ) b, integer ldb, integer info)

STPTRS

Purpose:

     STPTRS 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 REAL 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 REAL 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 stptrs.f.

  subroutine stpttf (character transr, character uplo, integer n, real, dimension( 0: * ) ap, real, dimension( 0: * ) arf, integer info)

STPTTF copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).

Purpose:

     STPTTF 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 REAL 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 REAL 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 stpttf.f.

  subroutine stpttr (character uplo, integer n, real, dimension( * ) ap, real, dimension( lda, * ) a, integer lda, integer info)

STPTTR copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).

Purpose:

     STPTTR 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 REAL 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 REAL 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 stpttr.f.

  subroutine strcon (character norm, character uplo, character diag, integer n, real, dimension( lda, * ) a, integer lda, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

STRCON

Purpose:

     STRCON 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 REAL 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 REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is REAL 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 strcon.f.

  subroutine strevc (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer info)

STREVC

Purpose:

     STREVC 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 SHSEQR.
     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 REAL 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 REAL 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 SHSEQR).
              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 REAL 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 SHSEQR).
              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 REAL 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 strevc.f.

  subroutine strevc3 (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer lwork, integer info)

STREVC3

Purpose:

     STREVC3 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 SHSEQR.
     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 REAL 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 REAL 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 SHSEQR).
              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 REAL 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 SHSEQR).
              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 REAL 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 strevc3.f.

  subroutine strexc (character compq, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldq, * ) q, integer ldq, integer ifst, integer ilst, real, dimension( * ) work, integer info)

STREXC

Purpose:

     STREXC 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 SHSEQR), 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 REAL 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 REAL 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 REAL 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 strexc.f.

  subroutine strrfs (character uplo, character trans, character diag, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

STRRFS

Purpose:

     STRRFS 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 STRTRS or some other
     means before entering this routine.  STRRFS 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 strrfs.f.

  subroutine strsen (character job, character compq, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) wr, real, dimension( * ) wi, integer m, real s, real sep, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

STRSEN

Purpose:

     STRSEN 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 SHSEQR), 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 REAL 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 REAL 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 REAL array, dimension (N)

    WI

              WI is REAL 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 REAL
              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 REAL
              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 REAL 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:

      STRSEN 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 312 of file strsen.f.

  subroutine strsna (character job, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) sep, integer mm, integer m, real, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)

STRSNA

Purpose:

     STRSNA 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 SHSEQR), 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 REAL 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 REAL 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
              SHSEIN or STREVC.
              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 REAL 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
              SHSEIN or STREVC.
              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 REAL 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 REAL 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 REAL 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 strsna.f.

  subroutine strti2 (character uplo, character diag, integer n, real, dimension( lda, * ) a, integer lda, integer info)

STRTI2 computes the inverse of a triangular matrix (unblocked algorithm).

Purpose:

     STRTI2 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 REAL 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 strti2.f.

  subroutine strtri (character uplo, character diag, integer n, real, dimension( lda, * ) a, integer lda, integer info)

STRTRI

Purpose:

     STRTRI 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 REAL 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 strtri.f.

  subroutine strtrs (character uplo, character trans, character diag, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer info)

STRTRS

Purpose:

     STRTRS 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 REAL 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 REAL 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 strtrs.f.

  subroutine strttf (character transr, character uplo, integer n, real, dimension( 0: lda-1, 0: * ) a, integer lda, real, dimension( 0: * ) arf, integer info)

STRTTF copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).

Purpose:

     STRTTF 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 REAL 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 REAL 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 strttf.f.

  subroutine strttp (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) ap, integer info)

STRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).

Purpose:

     STRTTP 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 REAL 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 REAL 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 strttp.f.

  subroutine stzrzf (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

STZRZF

Purpose:

     STZRZF 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 REAL 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 REAL array, dimension (M)
              The scalar factors of the elementary reflectors.

    WORK

              WORK is REAL 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 STZRZF,
      and tau(k) is the kth element of the array TAU.

Definition at line 150 of file stzrzf.f.

Author

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