complex16GEcomputational(3)

NAME
complex16GEcomputational - complex16

SYNOPSIS

  Functions
subroutine zgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
ZGEBAK
subroutine zgebal (job, n, a, lda, ilo, ihi, scale, info)
ZGEBAL
subroutine zgebd2 (m, n, a, lda, d, e, tauq, taup, work, info)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine zgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
ZGEBRD
subroutine zgecon (norm, n, a, lda, anorm, rcond, work, rwork, info)
ZGECON
subroutine zgeequ (m, n, a, lda, r, c, rowcnd, colcnd, amax, info)
ZGEEQU
subroutine zgeequb (m, n, a, lda, r, c, rowcnd, colcnd, amax, info)
ZGEEQUB
subroutine zgehd2 (n, ilo, ihi, a, lda, tau, work, info)
ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine zgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
ZGEHRD
subroutine zgelq2 (m, n, a, lda, tau, work, info)
ZGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine zgelqf (m, n, a, lda, tau, work, lwork, info)
ZGELQF
subroutine zgemqrt (side, trans, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info)
ZGEMQRT
subroutine zgeql2 (m, n, a, lda, tau, work, info)
ZGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
subroutine zgeqlf (m, n, a, lda, tau, work, lwork, info)
ZGEQLF
subroutine zgeqp3 (m, n, a, lda, jpvt, tau, work, lwork, rwork, info)
ZGEQP3
subroutine zgeqr2 (m, n, a, lda, tau, work, info)
ZGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
subroutine zgeqr2p (m, n, a, lda, tau, work, info)
ZGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
subroutine zgeqrf (m, n, a, lda, tau, work, lwork, info)
ZGEQRF
subroutine zgeqrfp (m, n, a, lda, tau, work, lwork, info)
ZGEQRFP
subroutine zgeqrt (m, n, nb, a, lda, t, ldt, work, info)
ZGEQRT
subroutine zgeqrt2 (m, n, a, lda, t, ldt, info)
ZGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
recursive subroutine zgeqrt3 (m, n, a, lda, t, ldt, info)
ZGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine zgerfs (trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZGERFS
subroutine zgerfsx (trans, equed, n, nrhs, a, lda, af, ldaf, ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
ZGERFSX
subroutine zgerq2 (m, n, a, lda, tau, work, info)
ZGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine zgerqf (m, n, a, lda, tau, work, lwork, info)
ZGERQF
subroutine zgesvj (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, cwork, lwork, rwork, lrwork, info)
ZGESVJ
subroutine zgetf2 (m, n, a, lda, ipiv, info)
ZGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
subroutine zgetrf (m, n, a, lda, ipiv, info)
ZGETRF
recursive subroutine zgetrf2 (m, n, a, lda, ipiv, info)
ZGETRF2
subroutine zgetri (n, a, lda, ipiv, work, lwork, info)
ZGETRI
subroutine zgetrs (trans, n, nrhs, a, lda, ipiv, b, ldb, info)
ZGETRS
subroutine zhgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
ZHGEQZ
subroutine zla_geamv (trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
double precision function zla_gercond_c (trans, n, a, lda, af, ldaf, ipiv, c, capply, info, work, rwork)
ZLA_GERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices.
double precision function zla_gercond_x (trans, n, a, lda, af, ldaf, ipiv, x, info, work, rwork)
ZLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general matrices.
subroutine zla_gerfsx_extended (prec_type, trans_type, n, nrhs, a, lda, af, ldaf, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, errs_n, errs_c, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
ZLA_GERFSX_EXTENDED
double precision function zla_gerpvgrw (n, ncols, a, lda, af, ldaf)
ZLA_GERPVGRW multiplies a square real matrix by a complex matrix.
recursive subroutine zlaqz0 (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)
ZLAQZ0
subroutine zlaqz1 (ilq, ilz, k, istartm, istopm, ihi, a, lda, b, ldb, nq, qstart, q, ldq, nz, zstart, z, ldz)
ZLAQZ1
recursive subroutine zlaqz2 (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alpha, beta, qc, ldqc, zc, ldzc, work, lwork, rwork, rec, info)
ZLAQZ2
subroutine zlaqz3 (ilschur, ilq, ilz, n, ilo, ihi, nshifts, nblock_desired, alpha, beta, a, lda, b, ldb, q, ldq, z, ldz, qc, ldqc, zc, ldzc, work, lwork, info)
ZLAQZ3
subroutine zlaunhr_col_getrfnp (m, n, a, lda, d, info)
ZLAUNHR_COL_GETRFNP
recursive subroutine zlaunhr_col_getrfnp2 (m, n, a, lda, d, info)
ZLAUNHR_COL_GETRFNP2
subroutine ztgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
ZTGEVC
subroutine ztgexc (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info)
ZTGEXC

Detailed Description

This is the group of complex16 computational functions for GE matrices

Function Documentation

  subroutine zgebak (character job, character side, integer n, integer ilo, integer ihi, double precision, dimension( * ) scale, integer m, complex*16, dimension( ldv, * ) v, integer ldv, integer info)

ZGEBAK

Purpose:

     ZGEBAK forms the right or left eigenvectors of a complex general
     matrix by backward transformation on the computed eigenvectors of the
     balanced matrix output by ZGEBAL.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies the type of backward transformation required:
              = 'N': do nothing, return immediately;
              = 'P': do backward transformation for permutation only;
              = 'S': do backward transformation for scaling only;
              = 'B': do backward transformations for both permutation and
                     scaling.
              JOB must be the same as the argument JOB supplied to ZGEBAL.

    SIDE

              SIDE is CHARACTER*1
              = 'R':  V contains right eigenvectors;
              = 'L':  V contains left eigenvectors.

    N

              N is INTEGER
              The number of rows of the matrix V.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              The integers ILO and IHI determined by ZGEBAL.
              1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

    SCALE

              SCALE is DOUBLE PRECISION array, dimension (N)
              Details of the permutation and scaling factors, as returned
              by ZGEBAL.

    M

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

    V

              V is COMPLEX*16 array, dimension (LDV,M)
              On entry, the matrix of right or left eigenvectors to be
              transformed, as returned by ZHSEIN or ZTREVC.
              On exit, V is overwritten by the transformed eigenvectors.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= 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 129 of file zgebak.f.

  subroutine zgebal (character job, integer n, complex*16, dimension( lda, * ) a, integer lda, integer ilo, integer ihi, double precision, dimension( * ) scale, integer info)

ZGEBAL

Purpose:

     ZGEBAL balances a general complex matrix A.  This involves, first,
     permuting A by a similarity transformation to isolate eigenvalues
     in the first 1 to ILO-1 and last IHI+1 to N elements on the
     diagonal; and second, applying a diagonal similarity transformation
     to rows and columns ILO to IHI to make the rows and columns as
     close in norm as possible.  Both steps are optional.
     Balancing may reduce the 1-norm of the matrix, and improve the
     accuracy of the computed eigenvalues and/or eigenvectors.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies the operations to be performed on A:
              = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(I) = 1.0
                      for i = 1,...,N;
              = 'P':  permute only;
              = 'S':  scale only;
              = 'B':  both permute and scale.

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the input matrix A.
              On exit,  A is overwritten by the balanced matrix.
              If JOB = 'N', A is not referenced.
              See Further Details.

    LDA

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI are set to INTEGER such that on exit
              A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
              If JOB = 'N' or 'S', ILO = 1 and IHI = N.

    SCALE

              SCALE is DOUBLE PRECISION array, dimension (N)
              Details of the permutations and scaling factors applied to
              A.  If P(j) is the index of the row and column interchanged
              with row and column j and D(j) is the scaling factor
              applied to row and column j, then
              SCALE(j) = P(j)    for j = 1,...,ILO-1
                       = D(j)    for j = ILO,...,IHI
                       = P(j)    for j = IHI+1,...,N.
              The order in which the interchanges are made is N to IHI+1,
              then 1 to ILO-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.

Further Details:

      The permutations consist of row and column interchanges which put
      the matrix in the form
                 ( T1   X   Y  )
         P A P = (  0   B   Z  )
                 (  0   0   T2 )
      where T1 and T2 are upper triangular matrices whose eigenvalues lie
      along the diagonal.  The column indices ILO and IHI mark the starting
      and ending columns of the submatrix B. Balancing consists of applying
      a diagonal similarity transformation inv(D) * B * D to make the
      1-norms of each row of B and its corresponding column nearly equal.
      The output matrix is
         ( T1     X*D          Y    )
         (  0  inv(D)*B*D  inv(D)*Z ).
         (  0      0           T2   )
      Information about the permutations P and the diagonal matrix D is
      returned in the vector SCALE.
      This subroutine is based on the EISPACK routine CBAL.
      Modified by Tzu-Yi Chen, Computer Science Division, University of
        California at Berkeley, USA

Definition at line 161 of file zgebal.f.

  subroutine zgebd2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tauq, complex*16, dimension( * ) taup, complex*16, dimension( * ) work, integer info)

ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

     ZGEBD2 reduces a complex general m by n matrix A to upper or lower
     real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
     If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters
    M

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

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n general matrix to be reduced.
              On exit,
              if m >= n, the diagonal and the first superdiagonal are
                overwritten with the upper bidiagonal matrix B; the
                elements below the diagonal, with the array TAUQ, represent
                the unitary matrix Q as a product of elementary
                reflectors, and the elements above the first superdiagonal,
                with the array TAUP, represent the unitary matrix P as
                a product of elementary reflectors;
              if m < n, the diagonal and the first subdiagonal are
                overwritten with the lower bidiagonal matrix B; the
                elements below the first subdiagonal, with the array TAUQ,
                represent the unitary matrix Q as a product of
                elementary reflectors, and the elements above the diagonal,
                with the array TAUP, represent the unitary matrix P as
                a product of elementary reflectors.
              See Further Details.

    LDA

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

    D

              D is DOUBLE PRECISION array, dimension (min(M,N))
              The diagonal elements of the bidiagonal matrix B:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION array, dimension (min(M,N)-1)
              The off-diagonal elements of the bidiagonal matrix B:
              if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
              if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

    TAUQ

              TAUQ is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix Q. See Further Details.

    TAUP

              TAUP is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix P. See Further Details.

    WORK

              WORK is COMPLEX*16 array, dimension (max(M,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 matrices Q and P are represented as products of elementary
      reflectors:
      If m >= n,
         Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
      Each H(i) and G(i) has the form:
         H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
      where tauq and taup are complex scalars, and v and u are complex
      vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
      A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
      A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      If m < n,
         Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
      Each H(i) and G(i) has the form:
         H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
      where tauq and taup are complex scalars, v and u are complex vectors;
      v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
      u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
      tauq is stored in TAUQ(i) and taup in TAUP(i).
      The contents of A on exit are illustrated by the following examples:
      m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
        (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
        (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
        (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
        (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
        (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
        (  v1  v2  v3  v4  v5 )
      where d and e denote diagonal and off-diagonal elements of B, vi
      denotes an element of the vector defining H(i), and ui an element of
      the vector defining G(i).

Definition at line 188 of file zgebd2.f.

  subroutine zgebrd (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tauq, complex*16, dimension( * ) taup, complex*16, dimension( * ) work, integer lwork, integer info)

ZGEBRD

Purpose:

     ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
     bidiagonal form B by a unitary transformation: Q**H * A * P = B.
     If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters
    M

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

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N general matrix to be reduced.
              On exit,
              if m >= n, the diagonal and the first superdiagonal are
                overwritten with the upper bidiagonal matrix B; the
                elements below the diagonal, with the array TAUQ, represent
                the unitary matrix Q as a product of elementary
                reflectors, and the elements above the first superdiagonal,
                with the array TAUP, represent the unitary matrix P as
                a product of elementary reflectors;
              if m < n, the diagonal and the first subdiagonal are
                overwritten with the lower bidiagonal matrix B; the
                elements below the first subdiagonal, with the array TAUQ,
                represent the unitary matrix Q as a product of
                elementary reflectors, and the elements above the diagonal,
                with the array TAUP, represent the unitary matrix P as
                a product of elementary reflectors.
              See Further Details.

    LDA

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

    D

              D is DOUBLE PRECISION array, dimension (min(M,N))
              The diagonal elements of the bidiagonal matrix B:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION array, dimension (min(M,N)-1)
              The off-diagonal elements of the bidiagonal matrix B:
              if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
              if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

    TAUQ

              TAUQ is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix Q. See Further Details.

    TAUP

              TAUP is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the unitary matrix P. See Further Details.

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= max(1,M,N).
              For optimum performance LWORK >= (M+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 matrices Q and P are represented as products of elementary
      reflectors:
      If m >= n,
         Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
      Each H(i) and G(i) has the form:
         H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
      where tauq and taup are complex scalars, and v and u are complex
      vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
      A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
      A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      If m < n,
         Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
      Each H(i) and G(i) has the form:
         H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
      where tauq and taup are complex scalars, and v and u are complex
      vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
      A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
      A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      The contents of A on exit are illustrated by the following examples:
      m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
        (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
        (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
        (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
        (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
        (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
        (  v1  v2  v3  v4  v5 )
      where d and e denote diagonal and off-diagonal elements of B, vi
      denotes an element of the vector defining H(i), and ui an element of
      the vector defining G(i).

Definition at line 203 of file zgebrd.f.

  subroutine zgecon (character norm, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision anorm, double precision rcond, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGECON

Purpose:

     ZGECON estimates the reciprocal of the condition number of a general
     complex matrix A, in either the 1-norm or the infinity-norm, using
     the LU factorization computed by ZGETRF.
     An estimate is obtained for norm(inv(A)), and 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.

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              The factors L and U from the factorization A = P*L*U
              as computed by ZGETRF.

    LDA

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

    ANORM

              ANORM is DOUBLE PRECISION
              If NORM = '1' or 'O', the 1-norm of the original matrix A.
              If NORM = 'I', the infinity-norm of the original matrix A.

    RCOND

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

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (2*N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 122 of file zgecon.f.

  subroutine zgeequ (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) r, double precision, dimension( * ) c, double precision rowcnd, double precision colcnd, double precision amax, integer info)

ZGEEQU

Purpose:

     ZGEEQU computes row and column scalings intended to equilibrate an
     M-by-N matrix A and reduce its condition number.  R returns the row
     scale factors and C the column scale factors, chosen to try to make
     the largest element in each row and column of the matrix B with
     elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
     R(i) and C(j) are restricted to be between SMLNUM = smallest safe
     number and BIGNUM = largest safe number.  Use of these scaling
     factors is not guaranteed to reduce the condition number of A but
     works well in practice.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              The M-by-N matrix whose equilibration factors are
              to be computed.

    LDA

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

    R

              R is DOUBLE PRECISION array, dimension (M)
              If INFO = 0 or INFO > M, R contains the row scale factors
              for A.

    C

              C is DOUBLE PRECISION array, dimension (N)
              If INFO = 0,  C contains the column scale factors for A.

    ROWCND

              ROWCND is DOUBLE PRECISION
              If INFO = 0 or INFO > M, ROWCND contains the ratio of the
              smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
              AMAX is neither too large nor too small, it is not worth
              scaling by R.

    COLCND

              COLCND is DOUBLE PRECISION
              If INFO = 0, COLCND contains the ratio of the smallest
              C(i) to the largest C(i).  If COLCND >= 0.1, it is not
              worth scaling by C.

    AMAX

              AMAX is DOUBLE PRECISION
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i,  and i is
                    <= M:  the i-th row of A is exactly zero
                    >  M:  the (i-M)-th column of A is exactly zero

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 138 of file zgeequ.f.

  subroutine zgeequb (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) r, double precision, dimension( * ) c, double precision rowcnd, double precision colcnd, double precision amax, integer info)

ZGEEQUB

Purpose:

     ZGEEQUB computes row and column scalings intended to equilibrate an
     M-by-N matrix A and reduce its condition number.  R returns the row
     scale factors and C the column scale factors, chosen to try to make
     the largest element in each row and column of the matrix B with
     elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
     the radix.
     R(i) and C(j) are restricted to be a power of the radix between
     SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
     of these scaling factors is not guaranteed to reduce the condition
     number of A but works well in practice.
     This routine differs from ZGEEQU by restricting the scaling factors
     to a power of the radix.  Barring over- and underflow, scaling by
     these factors introduces no additional rounding errors.  However, the
     scaled entries' magnitudes are no longer approximately 1 but lie
     between sqrt(radix) and 1/sqrt(radix).

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              The M-by-N matrix whose equilibration factors are
              to be computed.

    LDA

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

    R

              R is DOUBLE PRECISION array, dimension (M)
              If INFO = 0 or INFO > M, R contains the row scale factors
              for A.

    C

              C is DOUBLE PRECISION array, dimension (N)
              If INFO = 0,  C contains the column scale factors for A.

    ROWCND

              ROWCND is DOUBLE PRECISION
              If INFO = 0 or INFO > M, ROWCND contains the ratio of the
              smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
              AMAX is neither too large nor too small, it is not worth
              scaling by R.

    COLCND

              COLCND is DOUBLE PRECISION
              If INFO = 0, COLCND contains the ratio of the smallest
              C(i) to the largest C(i).  If COLCND >= 0.1, it is not
              worth scaling by C.

    AMAX

              AMAX is DOUBLE PRECISION
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i,  and i is
                    <= M:  the i-th row of A is exactly zero
                    >  M:  the (i-M)-th column of A is exactly zero

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 145 of file zgeequb.f.

  subroutine zgehd2 (integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

     ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
     by a unitary similarity transformation:  Q**H * A * Q = H .

Parameters
    N

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              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 ZGEBAL; otherwise they should be
              set to 1 and N respectively. See Further Details.
              1 <= ILO <= IHI <= max(1,N).

    A

              A is COMPLEX*16 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
              elements below the first subdiagonal, with the array TAU,
              represent the unitary 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,N).

    TAU

              TAU is COMPLEX*16 array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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:

      The matrix Q is represented as a product of (ihi-ilo) elementary
      reflectors
         Q = H(ilo) H(ilo+1) . . . H(ihi-1).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
      exit in A(i+2:ihi,i), and tau in TAU(i).
      The contents of A are illustrated by the following example, with
      n = 7, ilo = 2 and ihi = 6:
      on entry,                        on exit,
      ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
      (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
      (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
      (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
      (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
      (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
      (                         a )    (                          a )
      where a denotes an element of the original matrix A, h denotes a
      modified element of the upper Hessenberg matrix H, and vi denotes an
      element of the vector defining H(i).

Definition at line 148 of file zgehd2.f.

  subroutine zgehrd (integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGEHRD

Purpose:

     ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
     an unitary similarity transformation:  Q**H * A * Q = H .

Parameters
    N

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              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 ZGEBAL; otherwise they should be
              set to 1 and N respectively. See Further Details.
              1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

    A

              A is COMPLEX*16 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
              elements below the first subdiagonal, with the array TAU,
              represent the unitary 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,N).

    TAU

              TAU is COMPLEX*16 array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
              zero.

    WORK

              WORK is COMPLEX*16 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 >= max(1,N).
              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.

Further Details:

      The matrix Q is represented as a product of (ihi-ilo) elementary
      reflectors
         Q = H(ilo) H(ilo+1) . . . H(ihi-1).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
      exit in A(i+2:ihi,i), and tau in TAU(i).
      The contents of A are illustrated by the following example, with
      n = 7, ilo = 2 and ihi = 6:
      on entry,                        on exit,
      ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
      (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
      (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
      (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
      (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
      (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
      (                         a )    (                          a )
      where a denotes an element of the original matrix A, h denotes a
      modified element of the upper Hessenberg matrix H, and vi denotes an
      element of the vector defining H(i).
      This file is a slight modification of LAPACK-3.0's ZGEHRD
      subroutine incorporating improvements proposed by Quintana-Orti and
      Van de Geijn (2006). (See ZLAHR2.)

Definition at line 166 of file zgehrd.f.

  subroutine zgelq2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     ZGELQ2 computes an LQ factorization of a complex m-by-n matrix A:
        A = ( L 0 ) *  Q
     where:
        Q is a n-by-n orthogonal matrix;
        L is a lower-triangular m-by-m matrix;
        0 is a m-by-(n-m) zero matrix, if m < n.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the m by min(m,n) lower trapezoidal matrix L (L is
              lower triangular if m <= n); the elements above the diagonal,
              with the array TAU, represent the unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 array, dimension (M)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
      A(i,i+1:n), and tau in TAU(i).

Definition at line 128 of file zgelq2.f.

  subroutine zgelqf (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGELQF

Purpose:

     ZGELQF computes an LQ factorization of a complex M-by-N matrix A:
        A = ( L 0 ) *  Q
     where:
        Q is a N-by-N orthogonal matrix;
        L is a lower-triangular M-by-M matrix;
        0 is a M-by-(N-M) zero matrix, if M < N.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and below the diagonal of the array
              contain the m-by-min(m,n) lower trapezoidal matrix L (L is
              lower triangular if m <= n); the elements above the diagonal,
              with the array TAU, represent the unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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.

Further Details:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(k)**H . . . H(2)**H H(1)**H, where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i-1) = 0 and v(i) = 1; conjg(v(i+1:n)) is stored on exit in
      A(i,i+1:n), and tau in TAU(i).

Definition at line 142 of file zgelqf.f.

  subroutine zgemqrt (character side, character trans, integer m, integer n, integer k, integer nb, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldc, * ) c, integer ldc, complex*16, dimension( * ) work, integer info)

ZGEMQRT

Purpose:

     ZGEMQRT overwrites the general complex M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q C            C Q
     TRANS = 'C':    Q**H C            C Q**H
     where Q is a complex orthogonal matrix defined as the product of K
     elementary reflectors:
           Q = H(1) H(2) . . . H(K) = I - V T V**H
     generated using the compact WY representation as returned by ZGEQRT.
     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**H from the Left;
              = 'R': apply Q or Q**H from the Right.

    TRANS

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

    M

              M is INTEGER
              The number of rows of the matrix 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.

    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 ZGEQRT.

    V

              V is COMPLEX*16 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
              ZGEQRT in the first K columns of its array argument A.

    LDV

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

    T

              T is COMPLEX*16 array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by ZGEQRT, stored as a NB-by-N matrix.

    LDT

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

    C

              C is COMPLEX*16 array, dimension (LDC,N)
              On entry, the M-by-N matrix C.
              On exit, C is overwritten by Q C, Q**H C, C Q**H or C Q.

    LDC

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

    WORK

              WORK is COMPLEX*16 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.

Definition at line 166 of file zgemqrt.f.

  subroutine zgeql2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     ZGEQL2 computes a QL factorization of a complex m by n matrix A:
     A = Q * L.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n matrix A.
              On exit, if m >= n, the lower triangle of the subarray
              A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
              if m <= n, the elements on and below the (n-m)-th
              superdiagonal contain the m by n lower trapezoidal matrix L;
              the remaining elements, with the array TAU, represent the
              unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(k) . . . H(2) H(1), where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
      A(1:m-k+i-1,n-k+i), and tau in TAU(i).

Definition at line 122 of file zgeql2.f.

  subroutine zgeqlf (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGEQLF

Purpose:

     ZGEQLF computes a QL factorization of a complex M-by-N matrix A:
     A = Q * L.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if m >= n, the lower triangle of the subarray
              A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
              if m <= n, the elements on and below the (n-m)-th
              superdiagonal contain the M-by-N lower trapezoidal matrix L;
              the remaining elements, with the array TAU, represent the
              unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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 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(k) . . . H(2) H(1), where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
      A(1:m-k+i-1,n-k+i), and tau in TAU(i).

Definition at line 137 of file zgeqlf.f.

  subroutine zgeqp3 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) jpvt, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZGEQP3

Purpose:

     ZGEQP3 computes a QR factorization with column pivoting of a
     matrix A:  A*P = Q*R  using Level 3 BLAS.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the upper triangle of the array contains the
              min(M,N)-by-N upper trapezoidal matrix R; the elements below
              the diagonal, together with the array TAU, represent the
              unitary matrix Q as a product of min(M,N) elementary
              reflectors.

    LDA

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

    JPVT

              JPVT is INTEGER array, dimension (N)
              On entry, if JPVT(J).ne.0, the J-th column of A is permuted
              to the front of A*P (a leading column); if JPVT(J)=0,
              the J-th column of A is a free column.
              On exit, if JPVT(J)=K, then the J-th column of A*P was the
              the K-th column of A.

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors.

    WORK

              WORK is COMPLEX*16 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 >= N+1.
              For optimal 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.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (2*N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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 - tau * v * v**H
      where tau is a complex scalar, and v is a real/complex vector
      with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
      A(i+1:m,i), and tau in TAU(i).

Contributors:
    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

Definition at line 157 of file zgeqp3.f.

  subroutine zgeqr2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     ZGEQR2 computes a QR factorization of a complex m-by-n matrix A:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a m-by-m orthogonal matrix;
        R is an upper-triangular n-by-n matrix;
        0 is a (m-n)-by-n zero matrix, if m > n.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(m,n) by n upper trapezoidal matrix R (R is
              upper triangular if m >= n); the elements below the diagonal,
              with the array TAU, represent the unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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:

      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 - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
      and tau in TAU(i).

Definition at line 129 of file zgeqr2.f.

  subroutine zgeqr2p (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.

Purpose:

     ZGEQR2P computes a QR factorization of a complex m-by-n matrix A:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a m-by-m orthogonal matrix;
        R is an upper-triangular n-by-n matrix with nonnegative diagonal
        entries;
        0 is a (m-n)-by-n zero matrix, if m > n.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(m,n) by n upper trapezoidal matrix R (R is
              upper triangular if m >= n). The diagonal entries of R
              are real and nonnegative; the elements below the diagonal,
              with the array TAU, represent the unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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:

      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 - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
      and tau in TAU(i).
     See Lapack Working Note 203 for details

Definition at line 133 of file zgeqr2p.f.

  subroutine zgeqrf (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGEQRF

Purpose:

     ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a M-by-M orthogonal matrix;
        R is an upper-triangular N-by-N matrix;
        0 is a (M-N)-by-N zero matrix, if M > N.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(M,N)-by-N upper trapezoidal matrix R (R is
              upper triangular if m >= n); the elements below the diagonal,
              with the array TAU, represent the unitary matrix Q as a
              product of min(m,n) elementary reflectors (see Further
              Details).

    LDA

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

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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 >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
              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 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 - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
      and tau in TAU(i).

Definition at line 145 of file zgeqrf.f.

  subroutine zgeqrfp (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGEQRFP

Purpose:

     ZGEQR2P computes a QR factorization of a complex M-by-N matrix A:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a M-by-M orthogonal matrix;
        R is an upper-triangular N-by-N matrix with nonnegative diagonal
        entries;
        0 is a (M-N)-by-N zero matrix, if M > N.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(M,N)-by-N upper trapezoidal matrix R (R is
              upper triangular if m >= n). The diagonal entries of R
              are real and nonnegative; The elements below the diagonal,
              with the array TAU, represent the unitary matrix Q as a
              product of min(m,n) elementary reflectors (see Further
              Details).

    LDA

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

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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 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 - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
      and tau in TAU(i).
     See Lapack Working Note 203 for details

Definition at line 148 of file zgeqrfp.f.

  subroutine zgeqrt (integer m, integer n, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) work, integer info)

ZGEQRT

Purpose:

     ZGEQRT computes a blocked QR factorization of a complex M-by-N matrix A
     using the compact WY representation of Q.

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.

    NB

              NB is INTEGER
              The block size to be used in the blocked QR.  MIN(M,N) >= NB >= 1.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(M,N)-by-N upper trapezoidal matrix R (R is
              upper triangular if M >= N); the elements below the diagonal
              are the columns of V.

    LDA

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

    T

              T is COMPLEX*16 array, dimension (LDT,MIN(M,N))
              The upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.  See below
              for further details.

    LDT

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

    WORK

              WORK is COMPLEX*16 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 matrix V stores the elementary reflectors H(i) in the i-th column
      below the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1       )
                       ( v1  1    )
                       ( v1 v2  1 )
                       ( v1 v2 v3 )
                       ( v1 v2 v3 )
      where the vi's represent the vectors which define H(i), which are returned
      in the matrix A.  The 1's along the diagonal of V are not stored in A.
      Let K=MIN(M,N).  The number of blocks is B = ceiling(K/NB), where each
      block is of order NB except for the last block, which is of order
      IB = K - (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-K matrix T as
                   T = (T1 T2 ... TB).

Definition at line 140 of file zgeqrt.f.

  subroutine zgeqrt2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, integer info)

ZGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

     ZGEQRT2 computes a QR factorization of a complex M-by-N matrix A,
     using the compact WY representation of Q.

Parameters
    M

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

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the complex M-by-N matrix A.  On exit, the elements on and
              above the diagonal contain the N-by-N upper triangular matrix R; the
              elements below the diagonal are the columns of V.  See below for
              further details.

    LDA

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

    T

              T is COMPLEX*16 array, dimension (LDT,N)
              The N-by-N upper triangular factor of the block reflector.
              The elements on and above the diagonal contain the block
              reflector T; the elements below the diagonal are not used.
              See below for 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 matrix V stores the elementary reflectors H(i) in the i-th column
      below the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1       )
                       ( v1  1    )
                       ( v1 v2  1 )
                       ( v1 v2 v3 )
                       ( v1 v2 v3 )
      where the vi's represent the vectors which define H(i), which are returned
      in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
      block reflector H is then given by
                   H = I - V * T * V**H
      where V**H is the conjugate transpose of V.

Definition at line 126 of file zgeqrt2.f.

  recursive subroutine zgeqrt3 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, integer info)

ZGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

     ZGEQRT3 recursively computes a QR factorization of a complex M-by-N
     matrix A, using the compact WY representation of Q.
     Based on the algorithm of Elmroth and Gustavson,
     IBM J. Res. Develop. Vol 44 No. 4 July 2000.

Parameters
    M

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

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the complex M-by-N matrix A.  On exit, the elements on
              and above the diagonal contain the N-by-N upper triangular matrix R;
              the elements below the diagonal are the columns of V.  See below for
              further details.

    LDA

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

    T

              T is COMPLEX*16 array, dimension (LDT,N)
              The N-by-N upper triangular factor of the block reflector.
              The elements on and above the diagonal contain the block
              reflector T; the elements below the diagonal are not used.
              See below for 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 matrix V stores the elementary reflectors H(i) in the i-th column
      below the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1       )
                       ( v1  1    )
                       ( v1 v2  1 )
                       ( v1 v2 v3 )
                       ( v1 v2 v3 )
      where the vi's represent the vectors which define H(i), which are returned
      in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
      block reflector H is then given by
                   H = I - V * T * V**H
      where V**H is the conjugate transpose of V.
      For details of the algorithm, see Elmroth and Gustavson (cited above).

Definition at line 131 of file zgeqrt3.f.

  subroutine zgerfs (character trans, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGERFS

Purpose:

     ZGERFS improves the computed solution to a system of linear
     equations and provides error bounds and backward error estimates for
     the solution.

Parameters
    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)

    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 COMPLEX*16 array, dimension (LDA,N)
              The original N-by-N matrix A.

    LDA

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

    AF

              AF is COMPLEX*16 array, dimension (LDAF,N)
              The factors L and U from the factorization A = P*L*U
              as computed by ZGETRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from ZGETRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    B

              B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by ZGETRS.
              On exit, the improved solution matrix X.

    LDX

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

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N)

    INFO

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

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 184 of file zgerfs.f.

  subroutine zgerfsx (character trans, character equed, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, double precision, dimension( * ) r, double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx , * ) x, integer ldx, double precision rcond, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGERFSX

Purpose:

        ZGERFSX improves the computed solution to a system of linear
        equations and provides error bounds and backward error estimates
        for the solution.  In addition to normwise error bound, the code
        provides maximum componentwise error bound if possible.  See
        comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
        error bounds.
        The original system of linear equations may have been equilibrated
        before calling this routine, as described by arguments EQUED, R
        and C below. In this case, the solution and error bounds returned
        are for the original unequilibrated system.

         Some optional parameters are bundled in the PARAMS array.  These
         settings determine how refinement is performed, but often the
         defaults are acceptable.  If the defaults are acceptable, users
         can pass NPARAMS = 0 which prevents the source code from accessing
         the PARAMS argument.

Parameters
    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)

    EQUED

              EQUED is CHARACTER*1
         Specifies the form of equilibration that was done to A
         before calling this routine. This is needed to compute
         the solution and error bounds correctly.
           = 'N':  No equilibration
           = 'R':  Row equilibration, i.e., A has been premultiplied by
                   diag(R).
           = 'C':  Column equilibration, i.e., A has been postmultiplied
                   by diag(C).
           = 'B':  Both row and column equilibration, i.e., A has been
                   replaced by diag(R) * A * diag(C).
                   The right hand side B has been changed accordingly.

    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 COMPLEX*16 array, dimension (LDA,N)
         The original N-by-N matrix A.

    LDA

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

    AF

              AF is COMPLEX*16 array, dimension (LDAF,N)
         The factors L and U from the factorization A = P*L*U
         as computed by ZGETRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from ZGETRF; for 1<=i<=N, row i of the
         matrix was interchanged with row IPIV(i).

    R

              R is DOUBLE PRECISION array, dimension (N)
         The row scale factors for A.  If EQUED = 'R' or 'B', A is
         multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
         is not accessed.
         If R is accessed, each element of R should be a power of the radix
         to ensure a reliable solution and error estimates. Scaling by
         powers of the radix does not cause rounding errors unless the
         result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    C

              C is DOUBLE PRECISION array, dimension (N)
         The column scale factors for A.  If EQUED = 'C' or 'B', A is
         multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
         is not accessed.
         If C is accessed, each element of C should be a power of the radix
         to ensure a reliable solution and error estimates. Scaling by
         powers of the radix does not cause rounding errors unless the
         result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    B

              B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)
         On entry, the solution matrix X, as computed by ZGETRS.
         On exit, the improved solution matrix X.

    LDX

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

    RCOND

              RCOND is DOUBLE PRECISION
         Reciprocal scaled condition number.  This is an estimate of the
         reciprocal Skeel condition number of the matrix A after
         equilibration (if done).  If this is less than the machine
         precision (in particular, if it is zero), the matrix is singular
         to working precision.  Note that the error may still be small even
         if this number is very small and the matrix appears ill-
         conditioned.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
         Componentwise relative backward error.  This is 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).

    N_ERR_BNDS

              N_ERR_BNDS is INTEGER
         Number of error bounds to return for each right hand side
         and each type (normwise or componentwise).  See ERR_BNDS_NORM and
         ERR_BNDS_COMP below.

    ERR_BNDS_NORM

              ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         normwise relative error, which is defined as follows:
         Normwise relative error in the ith solution vector:
                 max_j (abs(XTRUE(j,i) - X(j,i)))
                ------------------------------
                      max_j abs(X(j,i))
         The array is indexed by the type of error information as described
         below. There currently are up to three pieces of information
         returned.
         The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_NORM(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * dlamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * dlamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated normwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * dlamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*A, where S scales each row by a power of the
                  radix so all absolute row sums of Z are approximately 1.
         See Lapack Working Note 165 for further details and extra
         cautions.

    ERR_BNDS_COMP

              ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         componentwise relative error, which is defined as follows:
         Componentwise relative error in the ith solution vector:
                        abs(XTRUE(j,i) - X(j,i))
                 max_j ----------------------
                             abs(X(j,i))
         The array is indexed by the right-hand side i (on which the
         componentwise relative error depends), and the type of error
         information as described below. There currently are up to three
         pieces of information returned for each right-hand side. If
         componentwise accuracy is not requested (PARAMS(3) = 0.0), then
         ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
         the first (:,N_ERR_BNDS) entries are returned.
         The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_COMP(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * dlamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * dlamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated componentwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * dlamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*(A*diag(x)), where x is the solution for the
                  current right-hand side and S scales each row of
                  A*diag(x) by a power of the radix so all absolute row
                  sums of Z are approximately 1.
         See Lapack Working Note 165 for further details and extra
         cautions.

    NPARAMS

              NPARAMS is INTEGER
         Specifies the number of parameters set in PARAMS.  If <= 0, the
         PARAMS array is never referenced and default values are used.

    PARAMS

              PARAMS is DOUBLE PRECISION array, dimension NPARAMS
         Specifies algorithm parameters.  If an entry is < 0.0, then
         that entry will be filled with default value used for that
         parameter.  Only positions up to NPARAMS are accessed; defaults
         are used for higher-numbered parameters.
           PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                refinement or not.
             Default: 1.0D+0
                = 0.0:  No refinement is performed, and no error bounds are
                        computed.
                = 1.0:  Use the double-precision refinement algorithm,
                        possibly with doubled-single computations if the
                        compilation environment does not support DOUBLE
                        PRECISION.
                  (other values are reserved for future use)
           PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                computations allowed for refinement.
             Default: 10
             Aggressive: Set to 100 to permit convergence using approximate
                         factorizations or factorizations other than LU. If
                         the factorization uses a technique other than
                         Gaussian elimination, the guarantees in
                         err_bnds_norm and err_bnds_comp may no longer be
                         trustworthy.
           PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                will attempt to find a solution with small componentwise
                relative error in the double-precision algorithm.  Positive
                is true, 0.0 is false.
             Default: 1.0 (attempt componentwise convergence)

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (2*N)

    INFO

              INFO is INTEGER
           = 0:  Successful exit. The solution to every right-hand side is
             guaranteed.
           < 0:  If INFO = -i, the i-th argument had an illegal value
           > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
             has been completed, but the factor U is exactly singular, so
             the solution and error bounds could not be computed. RCOND = 0
             is returned.
           = N+J: The solution corresponding to the Jth right-hand side is
             not guaranteed. The solutions corresponding to other right-
             hand sides K with K > J may not be guaranteed as well, but
             only the first such right-hand side is reported. If a small
             componentwise error is not requested (PARAMS(3) = 0.0) then
             the Jth right-hand side is the first with a normwise error
             bound that is not guaranteed (the smallest J such
             that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
             the Jth right-hand side is the first with either a normwise or
             componentwise error bound that is not guaranteed (the smallest
             J such that either ERR_BNDS_NORM(J,1) = 0.0 or
             ERR_BNDS_COMP(J,1) = 0.0). See the definition of
             ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
             about all of the right-hand sides check ERR_BNDS_NORM or
             ERR_BNDS_COMP.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 410 of file zgerfsx.f.

  subroutine zgerq2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     ZGERQ2 computes an RQ factorization of a complex m by n matrix A:
     A = R * Q.

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.

    A

              A is COMPLEX*16 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 TAU, represent the unitary 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).

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 array, dimension (M)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(1)**H H(2)**H . . . H(k)**H, where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
      exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).

Definition at line 122 of file zgerq2.f.

  subroutine zgerqf (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZGERQF

Purpose:

     ZGERQF computes an RQ factorization of a complex M-by-N matrix A:
     A = R * Q.

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.

    A

              A is COMPLEX*16 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 TAU, represent the
              unitary matrix Q as a product of min(m,n) elementary
              reflectors (see Further Details).

    LDA

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

    TAU

              TAU is COMPLEX*16 array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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 >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
              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.

Further Details:

      The matrix Q is represented as a product of elementary reflectors
         Q = H(1)**H H(2)**H . . . H(k)**H, where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on
      exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i).

Definition at line 138 of file zgerqf.f.

  subroutine zgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( n ) sva, integer mv, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( lwork ) cwork, integer lwork, double precision, dimension( lrwork ) rwork, integer lrwork, integer info)

ZGESVJ

Purpose:

     ZGESVJ computes the singular value decomposition (SVD) of a complex
     M-by-N matrix A, where M >= N. The SVD of A is written as
                                        [++]   [xx]   [x0]   [xx]
                  A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
                                        [++]   [xx]
     where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
     matrix, and V is an N-by-N unitary matrix. The diagonal elements
     of SIGMA are the singular values of A. The columns of U and V are the
     left and the right singular vectors of A, respectively.

Parameters
    JOBA

              JOBA is CHARACTER*1
              Specifies the structure of A.
              = 'L': The input matrix A is lower triangular;
              = 'U': The input matrix A is upper triangular;
              = 'G': The input matrix A is general M-by-N matrix, M >= N.

    JOBU

              JOBU is CHARACTER*1
              Specifies whether to compute the left singular vectors
              (columns of U):
              = 'U' or 'F': The left singular vectors corresponding to the nonzero
                     singular values are computed and returned in the leading
                     columns of A. See more details in the description of A.
                     The default numerical orthogonality threshold is set to
                     approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=DLAMCH('E').
              = 'C': Analogous to JOBU='U', except that user can control the
                     level of numerical orthogonality of the computed left
                     singular vectors. TOL can be set to TOL = CTOL*EPS, where
                     CTOL is given on input in the array WORK.
                     No CTOL smaller than ONE is allowed. CTOL greater
                     than 1 / EPS is meaningless. The option 'C'
                     can be used if M*EPS is satisfactory orthogonality
                     of the computed left singular vectors, so CTOL=M could
                     save few sweeps of Jacobi rotations.
                     See the descriptions of A and WORK(1).
              = 'N': The matrix U is not computed. However, see the
                     description of A.

    JOBV

              JOBV is CHARACTER*1
              Specifies whether to compute the right singular vectors, that
              is, the matrix V:
              = 'V' or 'J': the matrix V is computed and returned in the array V
              = 'A':  the Jacobi rotations are applied to the MV-by-N
                      array V. In other words, the right singular vector
                      matrix V is not computed explicitly; instead it is
                      applied to an MV-by-N matrix initially stored in the
                      first MV rows of V.
              = 'N':  the matrix V is not computed and the array V is not
                      referenced

    M

              M is INTEGER
              The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              If JOBU = 'U' .OR. JOBU = 'C':
                     If INFO = 0 :
                     RANKA orthonormal columns of U are returned in the
                     leading RANKA columns of the array A. Here RANKA <= N
                     is the number of computed singular values of A that are
                     above the underflow threshold DLAMCH('S'). The singular
                     vectors corresponding to underflowed or zero singular
                     values are not computed. The value of RANKA is returned
                     in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
                     descriptions of SVA and RWORK. The computed columns of U
                     are mutually numerically orthogonal up to approximately
                     TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
                     see the description of JOBU.
                     If INFO > 0,
                     the procedure ZGESVJ did not converge in the given number
                     of iterations (sweeps). In that case, the computed
                     columns of U may not be orthogonal up to TOL. The output
                     U (stored in A), SIGMA (given by the computed singular
                     values in SVA(1:N)) and V is still a decomposition of the
                     input matrix A in the sense that the residual
                     || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
              If JOBU = 'N':
                     If INFO = 0 :
                     Note that the left singular vectors are 'for free' in the
                     one-sided Jacobi SVD algorithm. However, if only the
                     singular values are needed, the level of numerical
                     orthogonality of U is not an issue and iterations are
                     stopped when the columns of the iterated matrix are
                     numerically orthogonal up to approximately M*EPS. Thus,
                     on exit, A contains the columns of U scaled with the
                     corresponding singular values.
                     If INFO > 0:
                     the procedure ZGESVJ did not converge in the given number
                     of iterations (sweeps).

    LDA

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

    SVA

              SVA is DOUBLE PRECISION array, dimension (N)
              On exit,
              If INFO = 0 :
              depending on the value SCALE = RWORK(1), we have:
                     If SCALE = ONE:
                     SVA(1:N) contains the computed singular values of A.
                     During the computation SVA contains the Euclidean column
                     norms of the iterated matrices in the array A.
                     If SCALE .NE. ONE:
                     The singular values of A are SCALE*SVA(1:N), and this
                     factored representation is due to the fact that some of the
                     singular values of A might underflow or overflow.
              If INFO > 0:
              the procedure ZGESVJ did not converge in the given number of
              iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.

    MV

              MV is INTEGER
              If JOBV = 'A', then the product of Jacobi rotations in ZGESVJ
              is applied to the first MV rows of V. See the description of JOBV.

    V

              V is COMPLEX*16 array, dimension (LDV,N)
              If JOBV = 'V', then V contains on exit the N-by-N matrix of
                             the right singular vectors;
              If JOBV = 'A', then V contains the product of the computed right
                             singular vector matrix and the initial matrix in
                             the array V.
              If JOBV = 'N', then V is not referenced.

    LDV

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

    CWORK

              CWORK is COMPLEX*16 array, dimension (max(1,LWORK))
              Used as workspace.
              If on entry LWORK = -1, then a workspace query is assumed and
              no computation is done; CWORK(1) is set to the minial (and optimal)
              length of CWORK.

    LWORK

              LWORK is INTEGER.
              Length of CWORK, LWORK >= M+N.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (max(6,LRWORK))
              On entry,
              If JOBU = 'C' :
              RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
                        The process stops if all columns of A are mutually
                        orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
                        It is required that CTOL >= ONE, i.e. it is not
                        allowed to force the routine to obtain orthogonality
                        below EPSILON.
              On exit,
              RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
                        are the computed singular values of A.
                        (See description of SVA().)
              RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
                        singular values.
              RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
                        values that are larger than the underflow threshold.
              RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
                        rotations needed for numerical convergence.
              RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
                        This is useful information in cases when ZGESVJ did
                        not converge, as it can be used to estimate whether
                        the output is still useful and for post festum analysis.
              RWORK(6) = the largest absolute value over all sines of the
                        Jacobi rotation angles in the last sweep. It can be
                        useful for a post festum analysis.
             If on entry LRWORK = -1, then a workspace query is assumed and
             no computation is done; RWORK(1) is set to the minial (and optimal)
             length of RWORK.

    LRWORK

             LRWORK is INTEGER
             Length of RWORK, LRWORK >= MAX(6,N).

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, then the i-th argument had an illegal value
              > 0:  ZGESVJ did not converge in the maximal allowed number
                    (NSWEEP=30) of sweeps. The output may still be useful.
                    See the description of RWORK.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
     rotations. In the case of underflow of the tangent of the Jacobi angle, a
     modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
     column interchanges of de Rijk [1]. The relative accuracy of the computed
     singular values and the accuracy of the computed singular vectors (in
     angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
     The condition number that determines the accuracy in the full rank case
     is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
     spectral condition number. The best performance of this Jacobi SVD
     procedure is achieved if used in an  accelerated version of Drmac and
     Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
     Some tuning parameters (marked with [TP]) are available for the
     implementer.
     The computational range for the nonzero singular values is the  machine
     number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
     denormalized singular values can be computed with the corresponding
     gradual loss of accurate digits.

Contributor:

      ============
      Zlatko Drmac (Zagreb, Croatia)

References:

     [1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
        singular value decomposition on a vector computer.
        SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
     [2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
     [3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
        value computation in floating point arithmetic.
        SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
     [4] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
        SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
        LAPACK Working note 169.
     [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
        SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
        LAPACK Working note 170.
     [6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
        QSVD, (H,K)-SVD computations.
        Department of Mathematics, University of Zagreb, 2008, 2015.

Bugs, examples and comments:

      ===========================
      Please report all bugs and send interesting test examples and comments to
      drmac@math.hr. Thank you.

Definition at line 349 of file zgesvj.f.

  subroutine zgetf2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

Purpose:

     ZGETF2 computes an LU factorization of a general m-by-n matrix A
     using partial pivoting with row interchanges.
     The factorization has the form
        A = P * L * U
     where P is a permutation matrix, L is lower triangular with unit
     diagonal elements (lower trapezoidal if m > n), and U is upper
     triangular (upper trapezoidal if m < n).
     This is the right-looking Level 2 BLAS version of the algorithm.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n matrix to be factored.
              On exit, the factors L and U from the factorization
              A = P*L*U; the unit diagonal elements of L are not stored.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (min(M,N))
              The pivot indices; for 1 <= i <= min(M,N), row i of the
              matrix was interchanged with row IPIV(i).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value
              > 0: if INFO = k, U(k,k) is exactly zero. The factorization
                   has been completed, but the factor U 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.

Definition at line 107 of file zgetf2.f.

  subroutine zgetrf (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZGETRF

Purpose:

     ZGETRF computes an LU factorization of a general M-by-N matrix A
     using partial pivoting with row interchanges.
     The factorization has the form
        A = P * L * U
     where P is a permutation matrix, L is lower triangular with unit
     diagonal elements (lower trapezoidal if m > n), and U is upper
     triangular (upper trapezoidal if m < n).
     This is the right-looking Level 3 BLAS version of the algorithm.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix to be factored.
              On exit, the factors L and U from the factorization
              A = P*L*U; the unit diagonal elements of L are not stored.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (min(M,N))
              The pivot indices; for 1 <= i <= min(M,N), row i of the
              matrix was interchanged with row IPIV(i).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                    has been completed, but the factor U 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.

Definition at line 107 of file zgetrf.f.

  recursive subroutine zgetrf2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZGETRF2

Purpose:

     ZGETRF2 computes an LU factorization of a general M-by-N matrix A
     using partial pivoting with row interchanges.
     The factorization has the form
        A = P * L * U
     where P is a permutation matrix, L is lower triangular with unit
     diagonal elements (lower trapezoidal if m > n), and U is upper
     triangular (upper trapezoidal if m < n).
     This is the recursive version of the algorithm. It divides
     the matrix into four submatrices:
            [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
        A = [ -----|----- ]  with n1 = min(m,n)/2
            [  A21 | A22  ]       n2 = n-n1
                                           [ A11 ]
     The subroutine calls itself to factor [ --- ],
                                           [ A12 ]
                     [ A12 ]
     do the swaps on [ --- ], solve A12, update A22,
                     [ A22 ]
     then calls itself to factor A22 and do the swaps on A21.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix to be factored.
              On exit, the factors L and U from the factorization
              A = P*L*U; the unit diagonal elements of L are not stored.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (min(M,N))
              The pivot indices; for 1 <= i <= min(M,N), row i of the
              matrix was interchanged with row IPIV(i).

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                    has been completed, but the factor U 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.

Definition at line 112 of file zgetrf2.f.

  subroutine zgetri (integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZGETRI

Purpose:

     ZGETRI computes the inverse of a matrix using the LU factorization
     computed by ZGETRF.
     This method inverts U and then computes inv(A) by solving the system
     inv(A)*L = inv(U) for inv(A).

Parameters
    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the factors L and U from the factorization
              A = P*L*U as computed by ZGETRF.
              On exit, if INFO = 0, the inverse of the original matrix A.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from ZGETRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO=0, then WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              For optimal performance LWORK >= N*NB, where NB is
              the optimal blocksize returned by ILAENV.
              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 INFO = i, U(i,i) is exactly zero; 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 113 of file zgetri.f.

  subroutine zgetrs (character trans, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZGETRS

Purpose:

     ZGETRS solves a system of linear equations
        A * X = B,  A**T * X = B,  or  A**H * X = B
     with a general N-by-N matrix A using the LU factorization computed
     by ZGETRF.

Parameters
    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)

    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 COMPLEX*16 array, dimension (LDA,N)
              The factors L and U from the factorization A = P*L*U
              as computed by ZGETRF.

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from ZGETRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    B

              B is COMPLEX*16 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 zgetrs.f.

  subroutine zhgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZHGEQZ

Purpose:

     ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
     where H is an upper Hessenberg matrix and T is upper triangular,
     using the single-shift QZ method.
     Matrix pairs of this type are produced by the reduction to
     generalized upper Hessenberg form of a complex matrix pair (A,B):
        A = Q1*H*Z1**H,  B = Q1*T*Z1**H,
     as computed by ZGGHRD.
     If JOB='S', then the Hessenberg-triangular pair (H,T) is
     also reduced to generalized Schur form,
        H = Q*S*Z**H,  T = Q*P*Z**H,
     where Q and Z are unitary matrices and S and P are upper triangular.
     Optionally, the unitary matrix Q from the generalized Schur
     factorization may be postmultiplied into an input matrix Q1, and the
     unitary matrix Z may be postmultiplied into an input matrix Z1.
     If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
     the matrix pair (A,B) to generalized Hessenberg form, then the output
     matrices Q1*Q and Z1*Z are the unitary factors from the generalized
     Schur factorization of (A,B):
        A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.
     To avoid overflow, eigenvalues of the matrix pair (H,T)
     (equivalently, of (A,B)) are computed as a pair of complex values
     (alpha,beta).  If beta is nonzero, lambda = alpha / beta is an
     eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
        A*x = lambda*B*x
     and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
     alternate form of the GNEP
        mu*A*y = B*y.
     The values of alpha and beta for the i-th eigenvalue can be read
     directly from the generalized Schur form:  alpha = S(i,i),
     beta = P(i,i).
     Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix
          Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
          pp. 241--256.

Parameters
    JOB

              JOB is CHARACTER*1
              = 'E': Compute eigenvalues only;
              = 'S': Computer eigenvalues and the Schur form.

    COMPQ

              COMPQ is CHARACTER*1
              = 'N': Left Schur vectors (Q) are not computed;
              = 'I': Q is initialized to the unit matrix and the matrix Q
                     of left Schur vectors of (H,T) is returned;
              = 'V': Q must contain a unitary matrix Q1 on entry and
                     the product Q1*Q is returned.

    COMPZ

              COMPZ is CHARACTER*1
              = 'N': Right Schur vectors (Z) are not computed;
              = 'I': Q is initialized to the unit matrix and the matrix Z
                     of right Schur vectors of (H,T) is returned;
              = 'V': Z must contain a unitary matrix Z1 on entry and
                     the product Z1*Z is returned.

    N

              N is INTEGER
              The order of the matrices H, T, Q, and Z.  N >= 0.

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI mark the rows and columns of H which are in
              Hessenberg form.  It is assumed that A is already upper
              triangular in rows and columns 1:ILO-1 and IHI+1:N.
              If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

    H

              H is COMPLEX*16 array, dimension (LDH, N)
              On entry, the N-by-N upper Hessenberg matrix H.
              On exit, if JOB = 'S', H contains the upper triangular
              matrix S from the generalized Schur factorization.
              If JOB = 'E', the diagonal of H matches that of S, but
              the rest of H is unspecified.

    LDH

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

    T

              T is COMPLEX*16 array, dimension (LDT, N)
              On entry, the N-by-N upper triangular matrix T.
              On exit, if JOB = 'S', T contains the upper triangular
              matrix P from the generalized Schur factorization.
              If JOB = 'E', the diagonal of T matches that of P, but
              the rest of T is unspecified.

    LDT

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

    ALPHA

              ALPHA is COMPLEX*16 array, dimension (N)
              The complex scalars alpha that define the eigenvalues of
              GNEP.  ALPHA(i) = S(i,i) in the generalized Schur
              factorization.

    BETA

              BETA is COMPLEX*16 array, dimension (N)
              The real non-negative scalars beta that define the
              eigenvalues of GNEP.  BETA(i) = P(i,i) in the generalized
              Schur factorization.
              Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
              represent the j-th eigenvalue of the matrix pair (A,B), in
              one of the forms lambda = alpha/beta or mu = beta/alpha.
              Since either lambda or mu may overflow, they should not,
              in general, be computed.

    Q

              Q is COMPLEX*16 array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
              reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPQ = 'I', the unitary matrix of left Schur
              vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
              left Schur vectors of (A,B).
              Not referenced if COMPQ = 'N'.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= 1.
              If COMPQ='V' or 'I', then LDQ >= N.

    Z

              Z is COMPLEX*16 array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
              reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPZ = 'I', the unitary matrix of right Schur
              vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
              right Schur vectors of (A,B).
              Not referenced if COMPZ = 'N'.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1.
              If COMPZ='V' or 'I', then LDZ >= N.

    WORK

              WORK is COMPLEX*16 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 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.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              = 1,...,N: the QZ iteration did not converge.  (H,T) is not
                         in Schur form, but ALPHA(i) and BETA(i),
                         i=INFO+1,...,N should be correct.
              = N+1,...,2*N: the shift calculation failed.  (H,T) is not
                         in Schur form, but ALPHA(i) and BETA(i),
                         i=INFO-N+1,...,N should be correct.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      We assume that complex ABS works as long as its value is less than
      overflow.

Definition at line 281 of file zhgeqz.f.

  subroutine zla_geamv (integer trans, integer m, integer n, double precision alpha, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) x, integer incx, double precision beta, double precision, dimension( * ) y, integer incy)

ZLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.

Purpose:

     ZLA_GEAMV  performs one of the matrix-vector operations
             y := alpha*abs(A)*abs(x) + beta*abs(y),
        or   y := alpha*abs(A)**T*abs(x) + beta*abs(y),
     where alpha and beta are scalars, x and y are vectors and A is an
     m by n matrix.
     This function is primarily used in calculating error bounds.
     To protect against underflow during evaluation, components in
     the resulting vector are perturbed away from zero by (N+1)
     times the underflow threshold.  To prevent unnecessarily large
     errors for block-structure embedded in general matrices,
     'symbolically' zero components are not perturbed.  A zero
     entry is considered 'symbolic' if all multiplications involved
     in computing that entry have at least one zero multiplicand.

Parameters
    TRANS

              TRANS is INTEGER
               On entry, TRANS specifies the operation to be performed as
               follows:
                 BLAS_NO_TRANS      y := alpha*abs(A)*abs(x) + beta*abs(y)
                 BLAS_TRANS         y := alpha*abs(A**T)*abs(x) + beta*abs(y)
                 BLAS_CONJ_TRANS    y := alpha*abs(A**T)*abs(x) + beta*abs(y)
               Unchanged on exit.

    M

              M is INTEGER
               On entry, M specifies the number of rows of the matrix A.
               M must be at least zero.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the number of columns of the matrix A.
               N must be at least zero.
               Unchanged on exit.

    ALPHA

              ALPHA is DOUBLE PRECISION
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    A

              A is COMPLEX*16 array, dimension ( LDA, n )
               Before entry, the leading m by n part of the array A must
               contain the matrix of coefficients.
               Unchanged on exit.

    LDA

              LDA is INTEGER
               On entry, LDA specifies the first dimension of A as declared
               in the calling (sub) program. LDA must be at least
               max( 1, m ).
               Unchanged on exit.

    X

              X is COMPLEX*16 array, dimension at least
               ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
               and at least
               ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
               Before entry, the incremented array X must contain the
               vector x.
               Unchanged on exit.

    INCX

              INCX is INTEGER
               On entry, INCX specifies the increment for the elements of
               X. INCX must not be zero.
               Unchanged on exit.

    BETA

              BETA is DOUBLE PRECISION
               On entry, BETA specifies the scalar beta. When BETA is
               supplied as zero then Y need not be set on input.
               Unchanged on exit.

    Y

              Y is DOUBLE PRECISION array, dimension
               ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
               and at least
               ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
               Before entry with BETA non-zero, the incremented array Y
               must contain the vector y. On exit, Y is overwritten by the
               updated vector y.

    INCY

              INCY is INTEGER
               On entry, INCY specifies the increment for the elements of
               Y. INCY must not be zero.
               Unchanged on exit.
      Level 2 Blas routine.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 173 of file zla_geamv.f.

  double precision function zla_gercond_c (character trans, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, double precision, dimension( * ) c, logical capply, integer info, complex*16, dimension( * ) work, double precision, dimension( * ) rwork)

ZLA_GERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices.

Purpose:

        ZLA_GERCOND_C computes the infinity norm condition number of
        op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector.

Parameters
    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)

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
         On entry, the N-by-N matrix A

    LDA

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

    AF

              AF is COMPLEX*16 array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by ZGETRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from the factorization A = P*L*U
         as computed by ZGETRF; row i of the matrix was interchanged
         with row IPIV(i).

    C

              C is DOUBLE PRECISION array, dimension (N)
         The vector C in the formula op(A) * inv(diag(C)).

    CAPPLY

              CAPPLY is LOGICAL
         If .TRUE. then access the vector C in the formula above.

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
         i > 0:  The ith argument is invalid.

    WORK

              WORK is COMPLEX*16 array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 140 of file zla_gercond_c.f.

  double precision function zla_gercond_x (character trans, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16, dimension( * ) x, integer info, complex*16, dimension( * ) work, double precision, dimension( * ) rwork)

ZLA_GERCOND_X computes the infinity norm condition number of op(A)*diag(x) for general matrices.

Purpose:

        ZLA_GERCOND_X computes the infinity norm condition number of
        op(A) * diag(X) where X is a COMPLEX*16 vector.

Parameters
    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)

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
         On entry, the N-by-N matrix A.

    LDA

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

    AF

              AF is COMPLEX*16 array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by ZGETRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from the factorization A = P*L*U
         as computed by ZGETRF; row i of the matrix was interchanged
         with row IPIV(i).

    X

              X is COMPLEX*16 array, dimension (N)
         The vector X in the formula op(A) * diag(X).

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
         i > 0:  The ith argument is invalid.

    WORK

              WORK is COMPLEX*16 array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 133 of file zla_gercond_x.f.

  subroutine zla_gerfsx_extended (integer prec_type, integer trans_type, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, logical colequ, double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldy, * ) y, integer ldy, double precision, dimension( * ) berr_out, integer n_norms, double precision, dimension( nrhs, * ) errs_n, double precision, dimension( nrhs, * ) errs_c, complex*16, dimension( * ) res, double precision, dimension( * ) ayb, complex*16, dimension( * ) dy, complex*16, dimension( * ) y_tail, double precision rcond, integer ithresh, double precision rthresh, double precision dz_ub, logical ignore_cwise, integer info)

ZLA_GERFSX_EXTENDED

Purpose:

     ZLA_GERFSX_EXTENDED improves the computed solution to a system of
     linear equations by performing extra-precise iterative refinement
     and provides error bounds and backward error estimates for the solution.
     This subroutine is called by ZGERFSX to perform iterative refinement.
     In addition to normwise error bound, the code provides maximum
     componentwise error bound if possible. See comments for ERRS_N
     and ERRS_C for details of the error bounds. Note that this
     subroutine is only responsible for setting the second fields of
     ERRS_N and ERRS_C.

Parameters
    PREC_TYPE

              PREC_TYPE is INTEGER
         Specifies the intermediate precision to be used in refinement.
         The value is defined by ILAPREC(P) where P is a CHARACTER and P
              = 'S':  Single
              = 'D':  Double
              = 'I':  Indigenous
              = 'X' or 'E':  Extra

    TRANS_TYPE

              TRANS_TYPE is INTEGER
         Specifies the transposition operation on A.
         The value is defined by ILATRANS(T) where T is a CHARACTER and T
              = 'N':  No transpose
              = 'T':  Transpose
              = 'C':  Conjugate transpose

    N

              N is INTEGER
         The number of linear equations, i.e., 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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
         On entry, the N-by-N matrix A.

    LDA

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

    AF

              AF is COMPLEX*16 array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by ZGETRF.

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from the factorization A = P*L*U
         as computed by ZGETRF; row i of the matrix was interchanged
         with row IPIV(i).

    COLEQU

              COLEQU is LOGICAL
         If .TRUE. then column equilibration was done to A before calling
         this routine. This is needed to compute the solution and error
         bounds correctly.

    C

              C is DOUBLE PRECISION array, dimension (N)
         The column scale factors for A. If COLEQU = .FALSE., C
         is not accessed. If C is input, each element of C should be a power
         of the radix to ensure a reliable solution and error estimates.
         Scaling by powers of the radix does not cause rounding errors unless
         the result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    B

              B is COMPLEX*16 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).

    Y

              Y is COMPLEX*16 array, dimension (LDY,NRHS)
         On entry, the solution matrix X, as computed by ZGETRS.
         On exit, the improved solution matrix Y.

    LDY

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

    BERR_OUT

              BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
         On exit, BERR_OUT(j) contains the componentwise relative backward
         error for right-hand-side j from the formula
             max(i) ( abs(RES(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. This is computed by ZLA_LIN_BERR.

    N_NORMS

              N_NORMS is INTEGER
         Determines which error bounds to return (see ERRS_N
         and ERRS_C).
         If N_NORMS >= 1 return normwise error bounds.
         If N_NORMS >= 2 return componentwise error bounds.

    ERRS_N

              ERRS_N is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         normwise relative error, which is defined as follows:
         Normwise relative error in the ith solution vector:
                 max_j (abs(XTRUE(j,i) - X(j,i)))
                ------------------------------
                      max_j abs(X(j,i))
         The array is indexed by the type of error information as described
         below. There currently are up to three pieces of information
         returned.
         The first index in ERRS_N(i,:) corresponds to the ith
         right-hand side.
         The second index in ERRS_N(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated normwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*A, where S scales each row by a power of the
                  radix so all absolute row sums of Z are approximately 1.
         This subroutine is only responsible for setting the second field
         above.
         See Lapack Working Note 165 for further details and extra
         cautions.

    ERRS_C

              ERRS_C is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         componentwise relative error, which is defined as follows:
         Componentwise relative error in the ith solution vector:
                        abs(XTRUE(j,i) - X(j,i))
                 max_j ----------------------
                             abs(X(j,i))
         The array is indexed by the right-hand side i (on which the
         componentwise relative error depends), and the type of error
         information as described below. There currently are up to three
         pieces of information returned for each right-hand side. If
         componentwise accuracy is not requested (PARAMS(3) = 0.0), then
         ERRS_C is not accessed.  If N_ERR_BNDS < 3, then at most
         the first (:,N_ERR_BNDS) entries are returned.
         The first index in ERRS_C(i,:) corresponds to the ith
         right-hand side.
         The second index in ERRS_C(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated componentwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*(A*diag(x)), where x is the solution for the
                  current right-hand side and S scales each row of
                  A*diag(x) by a power of the radix so all absolute row
                  sums of Z are approximately 1.
         This subroutine is only responsible for setting the second field
         above.
         See Lapack Working Note 165 for further details and extra
         cautions.

    RES

              RES is COMPLEX*16 array, dimension (N)
         Workspace to hold the intermediate residual.

    AYB

              AYB is DOUBLE PRECISION array, dimension (N)
         Workspace.

    DY

              DY is COMPLEX*16 array, dimension (N)
         Workspace to hold the intermediate solution.

    Y_TAIL

              Y_TAIL is COMPLEX*16 array, dimension (N)
         Workspace to hold the trailing bits of the intermediate solution.

    RCOND

              RCOND is DOUBLE PRECISION
         Reciprocal scaled condition number.  This is an estimate of the
         reciprocal Skeel condition number of the matrix A after
         equilibration (if done).  If this is less than the machine
         precision (in particular, if it is zero), the matrix is singular
         to working precision.  Note that the error may still be small even
         if this number is very small and the matrix appears ill-
         conditioned.

    ITHRESH

              ITHRESH is INTEGER
         The maximum number of residual computations allowed for
         refinement. The default is 10. For 'aggressive' set to 100 to
         permit convergence using approximate factorizations or
         factorizations other than LU. If the factorization uses a
         technique other than Gaussian elimination, the guarantees in
         ERRS_N and ERRS_C may no longer be trustworthy.

    RTHRESH

              RTHRESH is DOUBLE PRECISION
         Determines when to stop refinement if the error estimate stops
         decreasing. Refinement will stop when the next solution no longer
         satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
         the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
         default value is 0.5. For 'aggressive' set to 0.9 to permit
         convergence on extremely ill-conditioned matrices. See LAWN 165
         for more details.

    DZ_UB

              DZ_UB is DOUBLE PRECISION
         Determines when to start considering componentwise convergence.
         Componentwise convergence is only considered after each component
         of the solution Y is stable, which we define as the relative
         change in each component being less than DZ_UB. The default value
         is 0.25, requiring the first bit to be stable. See LAWN 165 for
         more details.

    IGNORE_CWISE

              IGNORE_CWISE is LOGICAL
         If .TRUE. then ignore componentwise convergence. Default value
         is .FALSE..

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
           < 0:  if INFO = -i, the ith argument to ZGETRS had an illegal
                 value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 390 of file zla_gerfsx_extended.f.

  double precision function zla_gerpvgrw (integer n, integer ncols, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf)

ZLA_GERPVGRW multiplies a square real matrix by a complex matrix.

Purpose:

     ZLA_GERPVGRW computes the reciprocal pivot growth factor
     norm(A)/norm(U). The 'max absolute element' norm is used. If this is
     much less than 1, the stability of the LU factorization of the
     (equilibrated) matrix A could be poor. This also means that the
     solution X, estimated condition numbers, and error bounds could be
     unreliable.

Parameters
    N

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

    NCOLS

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
         On entry, the N-by-N matrix A.

    LDA

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

    AF

              AF is COMPLEX*16 array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by ZGETRF.

    LDAF

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 98 of file zla_gerpvgrw.f.

  recursive subroutine zlaqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, complex*16, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex*16, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension( * ), intent(inout) beta, complex*16, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex*16, dimension( * ), intent(inout) work, integer, intent(in) lwork, double precision, dimension( * ), intent(out) rwork, integer, intent(in) rec, integer, intent(out) info)

ZLAQZ0

Purpose:

     ZLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
     where H is an upper Hessenberg matrix and T is upper triangular,
     using the double-shift QZ method.
     Matrix pairs of this type are produced by the reduction to
     generalized upper Hessenberg form of a real matrix pair (A,B):
        A = Q1*H*Z1**H,  B = Q1*T*Z1**H,
     as computed by ZGGHRD.
     If JOB='S', then the Hessenberg-triangular pair (H,T) is
     also reduced to generalized Schur form,
        H = Q*S*Z**H,  T = Q*P*Z**H,
     where Q and Z are unitary matrices, P and S are an upper triangular
     matrices.
     Optionally, the unitary matrix Q from the generalized Schur
     factorization may be postmultiplied into an input matrix Q1, and the
     unitary matrix Z may be postmultiplied into an input matrix Z1.
     If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
     the matrix pair (A,B) to generalized upper Hessenberg form, then the
     output matrices Q1*Q and Z1*Z are the unitary factors from the
     generalized Schur factorization of (A,B):
        A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.
     To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
     of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
     complex and beta real.
     If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
     generalized nonsymmetric eigenvalue problem (GNEP)
        A*x = lambda*B*x
     and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
     alternate form of the GNEP
        mu*A*y = B*y.
     Eigenvalues can be read directly from the generalized Schur
     form:
       alpha = S(i,i), beta = P(i,i).
     Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix
          Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
          pp. 241--256.
     Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ
          Algorithm with Aggressive Early Deflation', SIAM J. Numer.
          Anal., 29(2006), pp. 199--227.
     Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift,
          multipole rational QZ method with agressive early deflation'

Parameters
    WANTS

              WANTS is CHARACTER*1
              = 'E': Compute eigenvalues only;
              = 'S': Compute eigenvalues and the Schur form.

    WANTQ

              WANTQ is CHARACTER*1
              = 'N': Left Schur vectors (Q) are not computed;
              = 'I': Q is initialized to the unit matrix and the matrix Q
                     of left Schur vectors of (A,B) is returned;
              = 'V': Q must contain an unitary matrix Q1 on entry and
                     the product Q1*Q is returned.

    WANTZ

              WANTZ is CHARACTER*1
              = 'N': Right Schur vectors (Z) are not computed;
              = 'I': Z is initialized to the unit matrix and the matrix Z
                     of right Schur vectors of (A,B) is returned;
              = 'V': Z must contain an unitary matrix Z1 on entry and
                     the product Z1*Z is returned.

    N

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI mark the rows and columns of A which are in
              Hessenberg form.  It is assumed that A is already upper
              triangular in rows and columns 1:ILO-1 and IHI+1:N.
              If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

    A

              A is COMPLEX*16 array, dimension (LDA, N)
              On entry, the N-by-N upper Hessenberg matrix A.
              On exit, if JOB = 'S', A contains the upper triangular
              matrix S from the generalized Schur factorization.
              If JOB = 'E', the diagonal blocks of A match those of S, but
              the rest of A is unspecified.

    LDA

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

    B

              B is COMPLEX*16 array, dimension (LDB, N)
              On entry, the N-by-N upper triangular matrix B.
              On exit, if JOB = 'S', B contains the upper triangular
              matrix P from the generalized Schur factorization;
              If JOB = 'E', the diagonal blocks of B match those of P, but
              the rest of B is unspecified.

    LDB

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

    ALPHA

              ALPHA is COMPLEX*16 array, dimension (N)
              Each scalar alpha defining an eigenvalue
              of GNEP.

    BETA

              BETA is COMPLEX*16 array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = ALPHA(j) and
              beta = BETA(j) represent the j-th eigenvalue of the matrix
              pair (A,B), in one of the forms lambda = alpha/beta or
              mu = beta/alpha.  Since either lambda or mu may overflow,
              they should not, in general, be computed.

    Q

              Q is COMPLEX*16 array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the unitary matrix Q1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPQ = 'I', the unitary matrix of left Schur
              vectors of (A,B), and if COMPQ = 'V', the unitary matrix
              of left Schur vectors of (A,B).
              Not referenced if COMPQ = 'N'.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= 1.
              If COMPQ='V' or 'I', then LDQ >= N.

    Z

              Z is COMPLEX*16 array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the unitary matrix Z1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPZ = 'I', the unitary matrix of
              right Schur vectors of (H,T), and if COMPZ = 'V', the
              unitary matrix of right Schur vectors of (A,B).
              Not referenced if COMPZ = 'N'.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1.
              If COMPZ='V' or 'I', then LDZ >= N.

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N)

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,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.

    REC

              REC is INTEGER
                 REC indicates the current recursion level. Should be set
                 to 0 on first call.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              = 1,...,N: the QZ iteration did not converge.  (A,B) is not
                         in Schur form, but ALPHA(i) and
                         BETA(i), i=INFO+1,...,N should be correct.

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 280 of file zlaqz0.f.

  subroutine zlaqz1 (logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) k, integer, intent(in) istartm, integer, intent(in) istopm, integer, intent(in) ihi, complex*16, dimension( lda, * ) a, integer, intent(in) lda, complex*16, dimension( ldb, * ) b, integer, intent(in) ldb, integer, intent(in) nq, integer, intent(in) qstart, complex*16, dimension( ldq, * ) q, integer, intent(in) ldq, integer, intent(in) nz, integer, intent(in) zstart, complex*16, dimension( ldz, * ) z, integer, intent(in) ldz)

ZLAQZ1

Purpose:

          ZLAQZ1 chases a 1x1 shift bulge in a matrix pencil down a single position

Parameters
    ILQ

              ILQ is LOGICAL
                  Determines whether or not to update the matrix Q

    ILZ

              ILZ is LOGICAL
                  Determines whether or not to update the matrix Z

    K

              K is INTEGER
                  Index indicating the position of the bulge.
                  On entry, the bulge is located in
                  (A(k+1,k),B(k+1,k)).
                  On exit, the bulge is located in
                  (A(k+2,k+1),B(k+2,k+1)).

    ISTARTM

              ISTARTM is INTEGER

    ISTOPM

              ISTOPM is INTEGER
                  Updates to (A,B) are restricted to
                  (istartm:k+2,k:istopm). It is assumed
                  without checking that istartm <= k+1 and
                  k+2 <= istopm

    IHI

              IHI is INTEGER

    A

              A is COMPLEX*16 array, dimension (LDA,N)

    LDA

              LDA is INTEGER
                  The leading dimension of A as declared in
                  the calling procedure.

    B

              B is COMPLEX*16 array, dimension (LDB,N)

    LDB

              LDB is INTEGER
                  The leading dimension of B as declared in
                  the calling procedure.

    NQ

              NQ is INTEGER
                  The order of the matrix Q

    QSTART

              QSTART is INTEGER
                  Start index of the matrix Q. Rotations are applied
                  To columns k+2-qStart:k+3-qStart of Q.

    Q

              Q is COMPLEX*16 array, dimension (LDQ,NQ)

    LDQ

              LDQ is INTEGER
                  The leading dimension of Q as declared in
                  the calling procedure.

    NZ

              NZ is INTEGER
                  The order of the matrix Z

    ZSTART

              ZSTART is INTEGER
                  Start index of the matrix Z. Rotations are applied
                  To columns k+1-qStart:k+2-qStart of Z.

    Z

              Z is COMPLEX*16 array, dimension (LDZ,NZ)

    LDZ

              LDZ is INTEGER
                  The leading dimension of Q as declared in
                  the calling procedure.

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 171 of file zlaqz1.f.

  recursive subroutine zlaqz2 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nw, complex*16, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex*16, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex*16, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension( * ), intent(inout) beta, complex*16, dimension( ldqc, * ) qc, integer, intent(in) ldqc, complex*16, dimension( ldzc, * ) zc, integer, intent(in) ldzc, complex*16, dimension( * ) work, integer, intent(in) lwork, double precision, dimension( * ) rwork, integer, intent(in) rec, integer, intent(out) info)

ZLAQZ2

Purpose:

     ZLAQZ2 performs AED

Parameters
    ILSCHUR

              ILSCHUR is LOGICAL
                  Determines whether or not to update the full Schur form

    ILQ

              ILQ is LOGICAL
                  Determines whether or not to update the matrix Q

    ILZ

              ILZ is LOGICAL
                  Determines whether or not to update the matrix Z

    N

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI mark the rows and columns of (A,B) which
              are to be normalized

    NW

              NW is INTEGER
              The desired size of the deflation window.

    A

              A is COMPLEX*16 array, dimension (LDA, N)

    LDA

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

    B

              B is COMPLEX*16 array, dimension (LDB, N)

    LDB

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

    Q

              Q is COMPLEX*16 array, dimension (LDQ, N)

    LDQ

              LDQ is INTEGER

    Z

              Z is COMPLEX*16 array, dimension (LDZ, N)

    LDZ

              LDZ is INTEGER

    NS

              NS is INTEGER
              The number of unconverged eigenvalues available to
              use as shifts.

    ND

              ND is INTEGER
              The number of converged eigenvalues found.

    ALPHA

              ALPHA is COMPLEX*16 array, dimension (N)
              Each scalar alpha defining an eigenvalue
              of GNEP.

    BETA

              BETA is COMPLEX*16 array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = ALPHA(j) and
              beta = BETA(j) represent the j-th eigenvalue of the matrix
              pair (A,B), in one of the forms lambda = alpha/beta or
              mu = beta/alpha.  Since either lambda or mu may overflow,
              they should not, in general, be computed.

    QC

              QC is COMPLEX*16 array, dimension (LDQC, NW)

    LDQC

              LDQC is INTEGER

    ZC

              ZC is COMPLEX*16 array, dimension (LDZC, NW)

    LDZC

              LDZ is INTEGER

    WORK

              WORK is COMPLEX*16 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 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.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N)

    REC

              REC is INTEGER
                 REC indicates the current recursion level. Should be set
                 to 0 on first call.

    INFO

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

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 230 of file zlaqz2.f.

  subroutine zlaqz3 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nshifts, integer, intent(in) nblock_desired, complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension( * ), intent(inout) beta, complex*16, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex*16, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex*16, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex*16, dimension( ldqc, * ), intent(inout) qc, integer, intent(in) ldqc, complex*16, dimension( ldzc, * ), intent(inout) zc, integer, intent(in) ldzc, complex*16, dimension( * ), intent(inout) work, integer, intent(in) lwork, integer, intent(out) info)

ZLAQZ3

Purpose:

     ZLAQZ3 Executes a single multishift QZ sweep

Parameters
    ILSCHUR

              ILSCHUR is LOGICAL
                  Determines whether or not to update the full Schur form

    ILQ

              ILQ is LOGICAL
                  Determines whether or not to update the matrix Q

    ILZ

              ILZ is LOGICAL
                  Determines whether or not to update the matrix Z

    N

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER

    NSHIFTS

              NSHIFTS is INTEGER
              The desired number of shifts to use

    NBLOCK_DESIRED

              NBLOCK_DESIRED is INTEGER
              The desired size of the computational windows

    ALPHA

              ALPHA is COMPLEX*16 array. SR contains
              the alpha parts of the shifts to use.

    BETA

              BETA is COMPLEX*16 array. SS contains
              the scale of the shifts to use.

    A

              A is COMPLEX*16 array, dimension (LDA, N)

    LDA

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

    B

              B is COMPLEX*16 array, dimension (LDB, N)

    LDB

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

    Q

              Q is COMPLEX*16 array, dimension (LDQ, N)

    LDQ

              LDQ is INTEGER

    Z

              Z is COMPLEX*16 array, dimension (LDZ, N)

    LDZ

              LDZ is INTEGER

    QC

              QC is COMPLEX*16 array, dimension (LDQC, NBLOCK_DESIRED)

    LDQC

              LDQC is INTEGER

    ZC

              ZC is COMPLEX*16 array, dimension (LDZC, NBLOCK_DESIRED)

    LDZC

              LDZ is INTEGER

    WORK

              WORK is COMPLEX*16 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 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
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 204 of file zlaqz3.f.

  subroutine zlaunhr_col_getrfnp (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) d, integer info)

ZLAUNHR_COL_GETRFNP

Purpose:

     ZLAUNHR_COL_GETRFNP computes the modified LU factorization without
     pivoting of a complex general M-by-N matrix A. The factorization has
     the form:
         A - S = L * U,
     where:
        S is a m-by-n diagonal sign matrix with the diagonal D, so that
        D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
        as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
        i-1 steps of Gaussian elimination. This means that the diagonal
        element at each step of 'modified' Gaussian elimination is
        at least one in absolute value (so that division-by-zero not
        not possible during the division by the diagonal element);
        L is a M-by-N lower triangular matrix with unit diagonal elements
        (lower trapezoidal if M > N);
        and U is a M-by-N upper triangular matrix
        (upper trapezoidal if M < N).
     This routine is an auxiliary routine used in the Householder
     reconstruction routine ZUNHR_COL. In ZUNHR_COL, this routine is
     applied to an M-by-N matrix A with orthonormal columns, where each
     element is bounded by one in absolute value. With the choice of
     the matrix S above, one can show that the diagonal element at each
     step of Gaussian elimination is the largest (in absolute value) in
     the column on or below the diagonal, so that no pivoting is required
     for numerical stability [1].
     For more details on the Householder reconstruction algorithm,
     including the modified LU factorization, see [1].
     This is the blocked right-looking version of the algorithm,
     calling Level 3 BLAS to update the submatrix. To factorize a block,
     this routine calls the recursive routine ZLAUNHR_COL_GETRFNP2.
     [1] 'Reconstructing Householder vectors from tall-skinny QR',
         G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
         E. Solomonik, J. Parallel Distrib. Comput.,
         vol. 85, pp. 3-31, 2015.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix to be factored.
              On exit, the factors L and U from the factorization
              A-S=L*U; the unit diagonal elements of L are not stored.

    LDA

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

    D

              D is COMPLEX*16 array, dimension min(M,N)
              The diagonal elements of the diagonal M-by-N sign matrix S,
              D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
              only ( +1.0, 0.0 ) or (-1.0, 0.0 ).

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

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

Definition at line 145 of file zlaunhr_col_getrfnp.f.

  recursive subroutine zlaunhr_col_getrfnp2 (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) d, integer info)

ZLAUNHR_COL_GETRFNP2

Purpose:

     ZLAUNHR_COL_GETRFNP2 computes the modified LU factorization without
     pivoting of a complex general M-by-N matrix A. The factorization has
     the form:
         A - S = L * U,
     where:
        S is a m-by-n diagonal sign matrix with the diagonal D, so that
        D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
        as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
        i-1 steps of Gaussian elimination. This means that the diagonal
        element at each step of 'modified' Gaussian elimination is at
        least one in absolute value (so that division-by-zero not
        possible during the division by the diagonal element);
        L is a M-by-N lower triangular matrix with unit diagonal elements
        (lower trapezoidal if M > N);
        and U is a M-by-N upper triangular matrix
        (upper trapezoidal if M < N).
     This routine is an auxiliary routine used in the Householder
     reconstruction routine ZUNHR_COL. In ZUNHR_COL, this routine is
     applied to an M-by-N matrix A with orthonormal columns, where each
     element is bounded by one in absolute value. With the choice of
     the matrix S above, one can show that the diagonal element at each
     step of Gaussian elimination is the largest (in absolute value) in
     the column on or below the diagonal, so that no pivoting is required
     for numerical stability [1].
     For more details on the Householder reconstruction algorithm,
     including the modified LU factorization, see [1].
     This is the recursive version of the LU factorization algorithm.
     Denote A - S by B. The algorithm divides the matrix B into four
     submatrices:
            [  B11 | B12  ]  where B11 is n1 by n1,
        B = [ -----|----- ]        B21 is (m-n1) by n1,
            [  B21 | B22  ]        B12 is n1 by n2,
                                   B22 is (m-n1) by n2,
                                   with n1 = min(m,n)/2, n2 = n-n1.
     The subroutine calls itself to factor B11, solves for B21,
     solves for B12, updates B22, then calls itself to factor B22.
     For more details on the recursive LU algorithm, see [2].
     ZLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
     routine ZLAUNHR_COL_GETRFNP, which uses blocked code calling
     Level 3 BLAS to update the submatrix. However, ZLAUNHR_COL_GETRFNP2
     is self-sufficient and can be used without ZLAUNHR_COL_GETRFNP.
     [1] 'Reconstructing Householder vectors from tall-skinny QR',
         G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
         E. Solomonik, J. Parallel Distrib. Comput.,
         vol. 85, pp. 3-31, 2015.
     [2] 'Recursion leads to automatic variable blocking for dense linear
         algebra algorithms', F. Gustavson, IBM J. of Res. and Dev.,
         vol. 41, no. 6, pp. 737-755, 1997.

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.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix to be factored.
              On exit, the factors L and U from the factorization
              A-S=L*U; the unit diagonal elements of L are not stored.

    LDA

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

    D

              D is COMPLEX*16 array, dimension min(M,N)
              The diagonal elements of the diagonal M-by-N sign matrix S,
              D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
              only ( +1.0, 0.0 ) or (-1.0, 0.0 ).

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

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

Definition at line 166 of file zlaunhr_col_getrfnp2.f.

  subroutine ztgevc (character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( lds, * ) s, integer lds, complex*16, dimension( ldp, * ) p, integer ldp, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZTGEVC

Purpose:

     ZTGEVC computes some or all of the right and/or left eigenvectors of
     a pair of complex matrices (S,P), where S and P are upper triangular.
     Matrix pairs of this type are produced by the generalized Schur
     factorization of a complex matrix pair (A,B):
        A = Q*S*Z**H,  B = Q*P*Z**H
     as computed by ZGGHRD + ZHGEQZ.
     The right eigenvector x and the left eigenvector y of (S,P)
     corresponding to an eigenvalue w are defined by:
        S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
     where y**H denotes the conjugate tranpose of y.
     The eigenvalues are not input to this routine, but are computed
     directly from the diagonal elements of S and P.
     This routine returns the matrices X and/or Y of right and left
     eigenvectors of (S,P), or the products Z*X and/or Q*Y,
     where Z and Q are input matrices.
     If Q and Z are the unitary factors from the generalized Schur
     factorization of a matrix pair (A,B), then Z*X and Q*Y
     are the matrices of right and left eigenvectors of (A,B).

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,
                     specified by the logical array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY='S', SELECT specifies the eigenvectors to be
              computed.  The eigenvector corresponding to the j-th
              eigenvalue is computed if SELECT(j) = .TRUE..
              Not referenced if HOWMNY = 'A' or 'B'.

    N

              N is INTEGER
              The order of the matrices S and P.  N >= 0.

    S

              S is COMPLEX*16 array, dimension (LDS,N)
              The upper triangular matrix S from a generalized Schur
              factorization, as computed by ZHGEQZ.

    LDS

              LDS is INTEGER
              The leading dimension of array S.  LDS >= max(1,N).

    P

              P is COMPLEX*16 array, dimension (LDP,N)
              The upper triangular matrix P from a generalized Schur
              factorization, as computed by ZHGEQZ.  P must have real
              diagonal elements.

    LDP

              LDP is INTEGER
              The leading dimension of array P.  LDP >= max(1,N).

    VL

              VL is COMPLEX*16 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 unitary matrix Q
              of left Schur vectors returned by ZHGEQZ).
              On exit, if SIDE = 'L' or 'B', VL contains:
              if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
              if HOWMNY = 'B', the matrix Q*Y;
              if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
                          SELECT, stored consecutively in the columns of
                          VL, in the same order as their eigenvalues.
              Not referenced if SIDE = 'R'.

    LDVL

              LDVL is INTEGER
              The leading dimension of array VL.  LDVL >= 1, and if
              SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.

    VR

              VR is COMPLEX*16 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 unitary matrix Z
              of right Schur vectors returned by ZHGEQZ).
              On exit, if SIDE = 'R' or 'B', VR contains:
              if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
              if HOWMNY = 'B', the matrix Z*X;
              if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
                          SELECT, stored consecutively in the columns of
                          VR, in the same order as their eigenvalues.
              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 eigenvector occupies one column.

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (2*N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 217 of file ztgevc.f.

  subroutine ztgexc (logical wantq, logical wantz, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer ifst, integer ilst, integer info)

ZTGEXC

Purpose:

     ZTGEXC reorders the generalized Schur decomposition of a complex
     matrix pair (A,B), using an unitary equivalence transformation
     (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with
     row index IFST is moved to row ILST.
     (A, B) must be in generalized Schur canonical form, that is, A and
     B are both upper triangular.
     Optionally, the matrices Q and Z of generalized Schur vectors are
     updated.
            Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
            Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H

Parameters
    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.

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the upper triangular matrix A in the pair (A, B).
              On exit, the updated matrix A.

    LDA

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

    B

              B is COMPLEX*16 array, dimension (LDB,N)
              On entry, the upper triangular matrix B in the pair (A, B).
              On exit, the updated matrix B.

    LDB

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

    Q

              Q is COMPLEX*16 array, dimension (LDQ,N)
              On entry, if WANTQ = .TRUE., the unitary matrix Q.
              On exit, the updated matrix Q.
              If WANTQ = .FALSE., Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q. LDQ >= 1;
              If WANTQ = .TRUE., LDQ >= N.

    Z

              Z is COMPLEX*16 array, dimension (LDZ,N)
              On entry, if WANTZ = .TRUE., the unitary matrix Z.
              On exit, the updated matrix Z.
              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.

    IFST

              IFST is INTEGER

    ILST

              ILST is INTEGER
              Specify the reordering of the diagonal blocks of (A, B).
              The block with row index IFST is moved to row ILST, by a
              sequence of swapping between adjacent blocks.

    INFO

              INFO is INTEGER
               =0:  Successful exit.
               <0:  if INFO = -i, the i-th argument had an illegal value.
               =1:  The transformed matrix pair (A, B) would be too far
                    from generalized Schur form; the problem is ill-
                    conditioned. (A, B) 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.

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 198 of file ztgexc.f.

Author

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