doubleGEcomputational(3)

NAME
doubleGEcomputational - double

SYNOPSIS

  Functions
subroutine cgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
CGELQT
recursive subroutine cgelqt3 (M, N, A, LDA, T, LDT, INFO)
CGELQT3
subroutine cgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
CGEMLQT
subroutine dgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
DGEBAK
subroutine dgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
DGEBAL
subroutine dgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
subroutine dgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
DGEBRD
subroutine dgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DGECON
subroutine dgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQU
subroutine dgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
DGEEQUB
subroutine dgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)
DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine dgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
DGEHRD
subroutine dgelq2 (M, N, A, LDA, TAU, WORK, INFO)
DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGELQF
subroutine dgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
DGELQT
recursive subroutine dgelqt3 (M, N, A, LDA, T, LDT, INFO)
DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine dgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
DGEMLQT
subroutine dgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO)
DGEMQRT
subroutine dgeql2 (M, N, A, LDA, TAU, WORK, INFO)
DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQLF
subroutine dgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
DGEQP3
subroutine dgeqr2 (M, N, A, LDA, TAU, WORK, INFO)
DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgeqr2p (M, N, A, LDA, TAU, WORK, INFO)
DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
subroutine dgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
subroutine dgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRFP
subroutine dgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO)
DGEQRT
subroutine dgeqrt2 (M, N, A, LDA, T, LDT, INFO)
DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
recursive subroutine dgeqrt3 (M, N, A, LDA, T, LDT, INFO)
DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine dgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DGERFS
subroutine dgerfsx (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, IWORK, INFO)
DGERFSX
subroutine dgerq2 (M, N, A, LDA, TAU, WORK, INFO)
DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
subroutine dgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGERQF
subroutine dgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO)
DGESVJ
subroutine dgetf2 (M, N, A, LDA, IPIV, INFO)
DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
subroutine dgetrf (M, N, A, LDA, IPIV, INFO)
DGETRF
recursive subroutine dgetrf2 (M, N, A, LDA, IPIV, INFO)
DGETRF2
subroutine dgetri (N, A, LDA, IPIV, WORK, LWORK, INFO)
DGETRI
subroutine dgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
subroutine dhgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DHGEQZ
subroutine dla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
double precision function dla_gercond (TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
DLA_GERCOND estimates the Skeel condition number for a general matrix.
subroutine dla_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)
DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
double precision function dla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF)
DLA_GERPVGRW
subroutine dlaorhr_col_getrfnp (M, N, A, LDA, D, INFO)
DLAORHR_COL_GETRFNP
recursive subroutine dlaorhr_col_getrfnp2 (M, N, A, LDA, D, INFO)
DLAORHR_COL_GETRFNP2
recursive subroutine dlaqz0 (WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
DLAQZ0
subroutine dlaqz1 (A, LDA, B, LDB, SR1, SR2, SI, BETA1, BETA2, V)
DLAQZ1
subroutine dlaqz2 (ILQ, ILZ, K, ISTARTM, ISTOPM, IHI, A, LDA, B, LDB, NQ, QSTART, Q, LDQ, NZ, ZSTART, Z, LDZ)
DLAQZ2
recursive subroutine dlaqz3 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHAR, ALPHAI, BETA, QC, LDQC, ZC, LDZC, WORK, LWORK, REC, INFO)
DLAQZ3
subroutine dlaqz4 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS, NBLOCK_DESIRED, SR, SI, SS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK, LWORK, INFO)
DLAQZ4
subroutine dtgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
DTGEVC
subroutine dtgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
DTGEXC
subroutine sgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
SGELQT
recursive subroutine sgelqt3 (M, N, A, LDA, T, LDT, INFO)
SGELQT3
subroutine sgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
SGEMLQT
recursive subroutine slaqz0 (WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
SLAQZ0
subroutine slaqz1 (A, LDA, B, LDB, SR1, SR2, SI, BETA1, BETA2, V)
SLAQZ1
subroutine slaqz2 (ILQ, ILZ, K, ISTARTM, ISTOPM, IHI, A, LDA, B, LDB, NQ, QSTART, Q, LDQ, NZ, ZSTART, Z, LDZ)
SLAQZ2
recursive subroutine slaqz3 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHAR, ALPHAI, BETA, QC, LDQC, ZC, LDZC, WORK, LWORK, REC, INFO)
SLAQZ3
subroutine slaqz4 (ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSHIFTS, NBLOCK_DESIRED, SR, SI, SS, A, LDA, B, LDB, Q, LDQ, Z, LDZ, QC, LDQC, ZC, LDZC, WORK, LWORK, INFO)
SLAQZ4
subroutine zgelqt (M, N, MB, A, LDA, T, LDT, WORK, INFO)
ZGELQT
recursive subroutine zgelqt3 (M, N, A, LDA, T, LDT, INFO)
ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.
subroutine zgemlqt (SIDE, TRANS, M, N, K, MB, V, LDV, T, LDT, C, LDC, WORK, INFO)
ZGEMLQT

Detailed Description

This is the group of double computational functions for GE matrices

Function Documentation

  subroutine cgelqt (integer M, integer N, integer MB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( * ) WORK, integer INFO)

CGELQT

Purpose:

     CGELQT computes a blocked LQ 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.

    MB

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

    A

              A is COMPLEX 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
              are the rows of V.

    LDA

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

    T

              T is COMPLEX 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 >= MB.

    WORK

              WORK is COMPLEX array, dimension (MB*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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (         1 v3 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/MB), where each
      block is of order MB except for the last block, which is of order
      IB = K - (B-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-K matrix T as
                   T = (T1 T2 ... TB).

Definition at line 123 of file cgelqt.f.

  recursive subroutine cgelqt3 (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, integer INFO)

CGELQT3

Purpose:

     CGELQT3 recursively computes a LQ 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 array, dimension (LDA,N)
              On entry, the complex M-by-N matrix A.  On exit, the elements on and
              below the diagonal contain the N-by-N lower triangular matrix L; the
              elements above the diagonal are the rows 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 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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (     1  v3 v3 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**T
      where V**T is the transpose of V.
      For details of the algorithm, see Elmroth and Gustavson (cited above).

Definition at line 115 of file cgelqt3.f.

  subroutine cgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer MB, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK, integer INFO)

CGEMLQT

Purpose:

     CGEMLQT overwrites the general real 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 CGELQT.
     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.

    MB

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

    V

              V is COMPLEX array, dimension
                                   (LDV,M) if SIDE = 'L',
                                   (LDV,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              CGELQT in the first K rows of its array argument A.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,K).

    T

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

    LDT

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

    C

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 151 of file cgemlqt.f.

  subroutine dgebak (character JOB, character SIDE, integer N, integer ILO, integer IHI, double precision, dimension( * ) SCALE, integer M, double precision, dimension( ldv, * ) V, integer LDV, integer INFO)

DGEBAK

Purpose:

     DGEBAK forms the right or left eigenvectors of a real general matrix
     by backward transformation on the computed eigenvectors of the
     balanced matrix output by DGEBAL.

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 DGEBAL.

    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 DGEBAL.
              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 DGEBAL.

    M

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

    V

              V is DOUBLE PRECISION array, dimension (LDV,M)
              On entry, the matrix of right or left eigenvectors to be
              transformed, as returned by DHSEIN or DTREVC.
              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 128 of file dgebak.f.

  subroutine dgebal (character JOB, integer N, double precision, dimension( lda, * ) A, integer LDA, integer ILO, integer IHI, double precision, dimension( * ) SCALE, integer INFO)

DGEBAL

Purpose:

     DGEBAL balances a general real 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 DOUBLE PRECISION 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 integers 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 BALANC.
      Modified by Tzu-Yi Chen, Computer Science Division, University of
        California at Berkeley, USA

Definition at line 159 of file dgebal.f.

  subroutine dgebd2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( * ) WORK, integer INFO)

DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

     DGEBD2 reduces a real general m by n matrix A to upper or lower
     bidiagonal form B by an orthogonal transformation: Q**T * 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 DOUBLE PRECISION 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 orthogonal matrix Q as a product of elementary
                reflectors, and the elements above the first superdiagonal,
                with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of
                elementary reflectors, and the elements above the diagonal,
                with the array TAUP, represent the orthogonal 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 DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Q. See Further Details.

    TAUP

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

    WORK

              WORK is DOUBLE PRECISION 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**T  and G(i) = I - taup * u * u**T
      where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T
      where tauq and taup are real scalars, and v and u are real 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 dgebd2.f.

  subroutine dgebrd (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEBRD

Purpose:

     DGEBRD reduces a general real M-by-N matrix A to upper or lower
     bidiagonal form B by an orthogonal transformation: Q**T * 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 DOUBLE PRECISION 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 orthogonal matrix Q as a product of elementary
                reflectors, and the elements above the first superdiagonal,
                with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of
                elementary reflectors, and the elements above the diagonal,
                with the array TAUP, represent the orthogonal 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 DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Q. See Further Details.

    TAUP

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The 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**T  and G(i) = I - taup * u * u**T
      where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T
      where tauq and taup are real scalars, and v and u are real 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 dgebrd.f.

  subroutine dgecon (character NORM, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DGECON

Purpose:

     DGECON estimates the reciprocal of the condition number of a general
     real matrix A, in either the 1-norm or the infinity-norm, using
     the LU factorization computed by DGETRF.
     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 DOUBLE PRECISION array, dimension (LDA,N)
              The factors L and U from the factorization A = P*L*U
              as computed by DGETRF.

    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 DOUBLE PRECISION array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 122 of file dgecon.f.

  subroutine dgeequ (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO)

DGEEQU

Purpose:

     DGEEQU 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 DOUBLE PRECISION 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 137 of file dgeequ.f.

  subroutine dgeequb (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, integer INFO)

DGEEQUB

Purpose:

     DGEEQUB 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 DGEEQU 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 DOUBLE PRECISION 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 144 of file dgeequb.f.

  subroutine dgehd2 (integer N, integer ILO, integer IHI, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)

DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

     DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
     an orthogonal similarity transformation:  Q**T * 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 DGEBAL; otherwise they should be
              set to 1 and N respectively. See Further Details.
              1 <= ILO <= IHI <= max(1,N).

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the n by n general matrix to be reduced.
              On exit, the upper triangle and the first subdiagonal of A
              are overwritten with the upper Hessenberg matrix H, and the
              elements below the first subdiagonal, with the array TAU,
              represent the orthogonal matrix Q as a product of elementary
              reflectors. See Further Details.

    LDA

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

    TAU

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      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**T
      where tau is a real scalar, and v is a real 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 dgehd2.f.

  subroutine dgehrd (integer N, integer ILO, integer IHI, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEHRD

Purpose:

     DGEHRD reduces a real general matrix A to upper Hessenberg form H by
     an orthogonal similarity transformation:  Q**T * 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 DGEBAL; 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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the N-by-N general matrix to be reduced.
              On exit, the upper triangle and the first subdiagonal of A
              are overwritten with the upper Hessenberg matrix H, and the
              elements below the first subdiagonal, with the array TAU,
              represent the orthogonal matrix Q as a product of elementary
              reflectors. See Further Details.

    LDA

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

    TAU

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

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= 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**T
      where tau is a real scalar, and v is a real 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 DGEHRD
      subroutine incorporating improvements proposed by Quintana-Orti and
      Van de Geijn (2006). (See DLAHR2.)

Definition at line 166 of file dgehrd.f.

  subroutine dgelq2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)

DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     DGELQ2 computes an LQ factorization of a real 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 DOUBLE PRECISION 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 orthogonal matrix Q as a
              product of elementary reflectors (see Further Details).

    LDA

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

    TAU

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

    WORK

              WORK is DOUBLE PRECISION 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(2) H(1), where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(1:i-1) = 0 and v(i) = 1; 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 dgelq2.f.

  subroutine dgelqf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGELQF

Purpose:

     DGELQF computes an LQ factorization of a real 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 DOUBLE PRECISION 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 orthogonal matrix Q as a
              product of elementary reflectors (see Further Details).

    LDA

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

    TAU

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,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(2) H(1), where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(1:i-1) = 0 and v(i) = 1; 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 dgelqf.f.

  subroutine dgelqt (integer M, integer N, integer MB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integer INFO)

DGELQT

Purpose:

     DGELQT computes a blocked LQ factorization of a real 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.

    MB

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

    A

              A is DOUBLE PRECISION 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
              are the rows of V.

    LDA

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

    T

              T is DOUBLE PRECISION 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 >= MB.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MB*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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (         1 v3 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/MB), where each
      block is of order MB except for the last block, which is of order
      IB = K - (B-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-K matrix T as
                   T = (T1 T2 ... TB).

Definition at line 138 of file dgelqt.f.

  recursive subroutine dgelqt3 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO)

DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

     DGELQT3 recursively computes a LQ factorization of a real 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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the real M-by-N matrix A.  On exit, the elements on and
              below the diagonal contain the N-by-N lower triangular matrix L; the
              elements above the diagonal are the rows 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 DOUBLE PRECISION 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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (     1  v3 v3 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**T
      where V**T is the transpose of V.
      For details of the algorithm, see Elmroth and Gustavson (cited above).

Definition at line 130 of file dgelqt3.f.

  subroutine dgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer MB, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) WORK, integer INFO)

DGEMLQT

Purpose:

     DGEMLQT overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q C            C Q
     TRANS = 'T':   Q**T C            C Q**T
     where Q is a real orthogonal matrix defined as the product of K
     elementary reflectors:
           Q = H(1) H(2) . . . H(K) = I - V T V**T
     generated using the compact WY representation as returned by DGELQT.
     Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

Parameters
    SIDE

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

    TRANS

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

    M

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

    N

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

    K

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

    MB

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

    V

              V is DOUBLE PRECISION array, dimension
                                   (LDV,M) if SIDE = 'L',
                                   (LDV,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              DGELQT in the first K rows of its array argument A.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,K).

    T

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

    LDT

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

    C

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

    LDC

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

    WORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 166 of file dgemlqt.f.

  subroutine dgemqrt (character SIDE, character TRANS, integer M, integer N, integer K, integer NB, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) WORK, integer INFO)

DGEMQRT

Purpose:

     DGEMQRT overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q C            C Q
     TRANS = 'T':   Q**T C            C Q**T
     where Q is a real orthogonal matrix defined as the product of K
     elementary reflectors:
           Q = H(1) H(2) . . . H(K) = I - V T V**T
     generated using the compact WY representation as returned by DGEQRT.
     Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

Parameters
    SIDE

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

    TRANS

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

    M

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

    N

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

    K

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

    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 CGEQRT.

    V

              V is DOUBLE PRECISION array, dimension (LDV,K)
              The i-th column must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              CGEQRT 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 DOUBLE PRECISION array, dimension (LDT,K)
              The upper triangular factors of the block reflectors
              as returned by CGEQRT, stored as a NB-by-N matrix.

    LDT

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

    C

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

    LDC

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

    WORK

              WORK is DOUBLE PRECISION array. 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 dgemqrt.f.

  subroutine dgeql2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)

DGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     DGEQL2 computes a QL factorization of a real 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 DOUBLE PRECISION 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
              orthogonal matrix Q as a product of elementary reflectors
              (see Further Details).

    LDA

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

    TAU

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      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**T
      where tau is a real scalar, and v is a real 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 dgeql2.f.

  subroutine dgeqlf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEQLF

Purpose:

     DGEQLF computes a QL factorization of a real 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 DOUBLE PRECISION 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
              orthogonal matrix Q as a product of elementary reflectors
              (see Further Details).

    LDA

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

    TAU

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              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**T
      where tau is a real scalar, and v is a real 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 dgeqlf.f.

  subroutine dgeqp3 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEQP3

Purpose:

     DGEQP3 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 DOUBLE PRECISION 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
              orthogonal 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 DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= 3*N+1.
              For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB
              is the optimal blocksize.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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**T
      where tau is a real 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 150 of file dgeqp3.f.

  subroutine dgeqr2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)

DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     DGEQR2 computes a QR factorization of a real 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 DOUBLE PRECISION 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 orthogonal matrix Q as a
              product of elementary reflectors (see Further Details).

    LDA

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

    TAU

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      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**T
      where tau is a real scalar, and v is a real 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 dgeqr2.f.

  subroutine dgeqr2p (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)

DGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.

Purpose:

     DGEQR2P computes a QR factorization of a real 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 DOUBLE PRECISION 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
              nonnegative; the elements below the diagonal,
              with the array TAU, represent the orthogonal matrix Q as a
              product of elementary reflectors (see Further Details).

    LDA

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

    TAU

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      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**T
      where tau is a real scalar, and v is a real 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 dgeqr2p.f.

  subroutine dgeqrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEQRF

Purpose:

     DGEQRF computes a QR factorization of a real 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 DOUBLE PRECISION 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 orthogonal 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 DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              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**T
      where tau is a real scalar, and v is a real 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 144 of file dgeqrf.f.

  subroutine dgeqrfp (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGEQRFP

Purpose:

     DGEQR2P computes a QR factorization of a real 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 DOUBLE PRECISION 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 nonnegative; the elements below the diagonal,
              with the array TAU, represent the orthogonal 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 DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              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**T
      where tau is a real scalar, and v is a real 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 dgeqrfp.f.

  subroutine dgeqrt (integer M, integer N, integer NB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( * ) WORK, integer INFO)

DGEQRT

Purpose:

     DGEQRT computes a blocked QR factorization of a real 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB*N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The 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 dgeqrt.f.

  subroutine dgeqrt2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO)

DGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

     DGEQRT2 computes a QR factorization of a real 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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the real 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 DOUBLE PRECISION 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**T
      where V**T is the transpose of V.

Definition at line 126 of file dgeqrt2.f.

  recursive subroutine dgeqrt3 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO)

DGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

     DGEQRT3 recursively computes a QR factorization of a real 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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the real 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 DOUBLE PRECISION 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**T
      where V**T is the transpose of V.
      For details of the algorithm, see Elmroth and Gustavson (cited above).

Definition at line 131 of file dgeqrt3.f.

  subroutine dgerfs (character TRANS, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DGERFS

Purpose:

     DGERFS 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 = 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
              The factors L and U from the factorization A = P*L*U
              as computed by DGETRF.

    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 DGETRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

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

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by DGETRS.
              On exit, the improved solution matrix X.

    LDX

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

    FERR

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

    BERR

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N)

    IWORK

              IWORK is INTEGER array, dimension (N)

    INFO

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

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 183 of file dgerfs.f.

  subroutine dgerfsx (character TRANS, character EQUED, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, 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, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DGERFSX

Purpose:

        DGERFSX 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 = 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
         The factors L and U from the factorization A = P*L*U
         as computed by DGETRF.

    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 DGETRF; 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 DOUBLE PRECISION array, dimension (LDB,NRHS)
         The right hand side matrix B.

    LDB

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

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)
         On entry, the solution matrix X, as computed by DGETRS.
         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 DOUBLE PRECISION array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (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 dgerfsx.f.

  subroutine dgerq2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO)

DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

     DGERQ2 computes an RQ factorization of a real 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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the m by n matrix A.
              On exit, if m <= n, the upper triangle of the subarray
              A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
              if m >= n, the elements on and above the (m-n)-th subdiagonal
              contain the m by n upper trapezoidal matrix R; the remaining
              elements, with the array TAU, represent the orthogonal matrix
              Q as a product of elementary reflectors (see Further
              Details).

    LDA

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

    TAU

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

    WORK

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

Definition at line 122 of file dgerq2.f.

  subroutine dgerqf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGERQF

Purpose:

     DGERQF computes an RQ factorization of a real 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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if m <= n, the upper triangle of the subarray
              A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
              if m >= n, the elements on and above the (m-n)-th subdiagonal
              contain the M-by-N upper trapezoidal matrix R;
              the remaining elements, with the array TAU, represent the
              orthogonal 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 DOUBLE PRECISION array, dimension (min(M,N))
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,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(1) H(2) . . . H(k), where k = min(m,n).
      Each H(i) has the form
         H(i) = I - tau * v * v**T
      where tau is a real scalar, and v is a real vector with
      v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
      A(m-k+i,1:n-k+i-1), and tau in TAU(i).

Definition at line 137 of file dgerqf.f.

  subroutine dgesvj (character*1 JOBA, character*1 JOBU, character*1 JOBV, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( n ) SVA, integer MV, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( lwork ) WORK, integer LWORK, integer INFO)

DGESVJ

Purpose:

     DGESVJ computes the singular value decomposition (SVD) of a real
     M-by-N matrix A, where M >= N. The SVD of A is written as
                                        [++]   [xx]   [x0]   [xx]
                  A = U * SIGMA * V^t,  [++] = [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 orthogonal 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.
     DGESVJ can sometimes compute tiny singular values and their singular vectors much
     more accurately than other SVD routines, see below under Further Details.

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': 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=DSQRT(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':  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 DOUBLE PRECISION 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 WORK as RANKA=NINT(WORK(2)). Also see the
                     descriptions of SVA and WORK. The computed columns of U
                     are mutually numerically orthogonal up to approximately
                     TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
                     see the description of JOBU.
                     If INFO > 0 :
                     the procedure DGESVJ 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^T||_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 DGESVJ 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 = WORK(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 DGESVJ 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 DGESVJ
              is applied to the first MV rows of V. See the description of JOBV.

    V

              V is DOUBLE PRECISION 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) .

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
              On entry :
              If JOBU = 'C' :
              WORK(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 EPS.
              On exit :
              WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
                        are the computed singular values of A.
                        (See description of SVA().)
              WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
                        singular values.
              WORK(3) = NINT(WORK(3)) is the number of the computed singular
                        values that are larger than the underflow threshold.
              WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
                        rotations needed for numerical convergence.
              WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
                        This is useful information in cases when DGESVJ did
                        not converge, as it can be used to estimate whether
                        the output is still useful and for post festum analysis.
              WORK(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.

    LWORK

              LWORK is INTEGER
              length of WORK, WORK >= MAX(6,M+N)

    INFO

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

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. The rotations are implemented as fast scaled rotations of
      Anda and Park [1]. In the case of underflow of the Jacobi angle, a
      modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
      column interchanges of de Rijk [2]. 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 [3].
      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 [5,6], and it is the kernel routine in the SIGMA library [7].
      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.

Contributors:

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

References:

     [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
         SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
     [2] 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.
     [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
     [4] 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.
     [5] 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.
     [6] 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.
     [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
         QSVD, (H,K)-SVD computations.
         Department of Mathematics, University of Zagreb, 2008.

Bugs, examples and comments:

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

Definition at line 335 of file dgesvj.f.

  subroutine dgetf2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

DGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

Purpose:

     DGETF2 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 DOUBLE PRECISION 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 dgetf2.f.

  subroutine dgetrf (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

DGETRF DGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm

DGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

Purpose:

     DGETRF 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 DOUBLE PRECISION 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.

Purpose:

 DGETRF 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 left-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 DOUBLE PRECISION 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.

Date
    December 2016

Purpose:

 DGETRF 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 code implements an iterative version of Sivan Toledo's recursive
 LU algorithm[1].  For square matrices, this iterative versions should
 be within a factor of two of the optimum number of memory transfers.
 The pattern is as follows, with the large blocks of U being updated
 in one call to DTRSM, and the dotted lines denoting sections that
 have had all pending permutations applied:
  1 2 3 4 5 6 7 8
 +-+-+---+-------+------
 | |1|   |       |
 |.+-+ 2 |       |
 | | |   |       |
 |.|.+-+-+   4   |
 | | | |1|       |
 | | |.+-+       |
 | | | | |       |
 |.|.|.|.+-+-+---+  8
 | | | | | |1|   |
 | | | | |.+-+ 2 |
 | | | | | | |   |
 | | | | |.|.+-+-+
 | | | | | | | |1|
 | | | | | | |.+-+
 | | | | | | | | |
 |.|.|.|.|.|.|.|.+-----
 | | | | | | | | |
 The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
 the binary expansion of the current column.  Each Schur update is
 applied as soon as the necessary portion of U is available.
 [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
 Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
 1065-1081. http://dx.doi.org/10.1137/S0895479896297744

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 DOUBLE PRECISION 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.

Date
    December 2016

Definition at line 107 of file dgetrf.f.

  recursive subroutine dgetrf2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

DGETRF2

Purpose:

     DGETRF2 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 DOUBLE PRECISION 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 dgetrf2.f.

  subroutine dgetri (integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGETRI

Purpose:

     DGETRI computes the inverse of a matrix using the LU factorization
     computed by DGETRF.
     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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the factors L and U from the factorization
              A = P*L*U as computed by DGETRF.
              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 DGETRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    WORK

              WORK is DOUBLE PRECISION 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 dgetri.f.

  subroutine dgetrs (character TRANS, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DGETRS

Purpose:

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

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**T* X = B  (Conjugate transpose = 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 DOUBLE PRECISION array, dimension (LDA,N)
              The factors L and U from the factorization A = P*L*U
              as computed by DGETRF.

    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 DGETRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    B

              B is DOUBLE PRECISION array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 120 of file dgetrs.f.

  subroutine dhgeqz (character JOB, character COMPQ, character COMPZ, integer N, integer ILO, integer IHI, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( * ) ALPHAR, double precision, dimension( * ) ALPHAI, double precision, dimension( * ) BETA, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DHGEQZ

Purpose:

     DHGEQZ 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**T,  B = Q1*T*Z1**T,
     as computed by DGGHRD.
     If JOB='S', then the Hessenberg-triangular pair (H,T) is
     also reduced to generalized Schur form,
        H = Q*S*Z**T,  T = Q*P*Z**T,
     where Q and Z are orthogonal matrices, P is an upper triangular
     matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
     diagonal blocks.
     The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
     (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
     eigenvalues.
     Additionally, the 2-by-2 upper triangular diagonal blocks of P
     corresponding to 2-by-2 blocks of S are reduced to positive diagonal
     form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
     P(j,j) > 0, and P(j+1,j+1) > 0.
     Optionally, the orthogonal matrix Q from the generalized Schur
     factorization may be postmultiplied into an input matrix Q1, and the
     orthogonal matrix Z may be postmultiplied into an input matrix Z1.
     If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
     the matrix pair (A,B) to generalized upper Hessenberg form, then the
     output matrices Q1*Q and Z1*Z are the orthogonal factors from the
     generalized Schur factorization of (A,B):
        A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
     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.
     Real 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.

Parameters
    JOB

              JOB is CHARACTER*1
              = 'E': Compute eigenvalues only;
              = 'S': Compute 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 an orthogonal 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': Z is initialized to the unit matrix and the matrix Z
                     of right Schur vectors of (H,T) is returned;
              = 'V': Z must contain an orthogonal matrix Z1 on entry and
                     the product Z1*Z is returned.

    N

              N is INTEGER
              The order of the matrices 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 DOUBLE PRECISION array, dimension (LDH, N)
              On entry, the N-by-N upper Hessenberg matrix H.
              On exit, if JOB = 'S', H contains the upper quasi-triangular
              matrix S from the generalized Schur factorization.
              If JOB = 'E', the diagonal blocks of H match those 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 DOUBLE PRECISION 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;
              2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
              are reduced to positive diagonal form, i.e., if H(j+1,j) is
              non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
              T(j+1,j+1) > 0.
              If JOB = 'E', the diagonal blocks of T match those of P, but
              the rest of T is unspecified.

    LDT

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

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)
              The real parts of each scalar alpha defining an eigenvalue
              of GNEP.

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)
              The imaginary parts of each scalar alpha defining an
              eigenvalue of GNEP.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 DOUBLE PRECISION array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
              vectors of (H,T), and if COMPQ = 'V', the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPZ = 'I', the orthogonal matrix of
              right Schur vectors of (H,T), and if COMPZ = 'V', the
              orthogonal 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              If 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
              = 1,...,N: the QZ iteration did not converge.  (H,T) is not
                         in Schur form, but ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(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:

      Iteration counters:
      JITER  -- counts iterations.
      IITER  -- counts iterations run since ILAST was last
                changed.  This is therefore reset only when a 1-by-1 or
                2-by-2 block deflates off the bottom.

Definition at line 301 of file dhgeqz.f.

  subroutine dla_geamv (integer TRANS, integer M, integer N, double precision ALPHA, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)

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

Purpose:

     DLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension
               ( 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 at least
               ( 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 172 of file dla_geamv.f.

  double precision function dla_gercond (character TRANS, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE, double precision, dimension( * ) C, integer INFO, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK)

DLA_GERCOND estimates the Skeel condition number for a general matrix.

Purpose:

        DLA_GERCOND estimates the Skeel condition number of op(A) * op2(C)
        where op2 is determined by CMODE as follows
        CMODE =  1    op2(C) = C
        CMODE =  0    op2(C) = I
        CMODE = -1    op2(C) = inv(C)
        The Skeel condition number cond(A) = norminf( |inv(A)||A| )
        is computed by computing scaling factors R such that
        diag(R)*A*op2(C) is row equilibrated and computing the standard
        infinity-norm condition number.

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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by DGETRF.

    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 DGETRF; row i of the matrix was interchanged
         with row IPIV(i).

    CMODE

              CMODE is INTEGER
         Determines op2(C) in the formula op(A) * op2(C) as follows:
         CMODE =  1    op2(C) = C
         CMODE =  0    op2(C) = I
         CMODE = -1    op2(C) = inv(C)

    C

              C is DOUBLE PRECISION array, dimension (N)
         The vector C in the formula op(A) * op2(C).

    INFO

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

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N).
         Workspace.

    IWORK

              IWORK is INTEGER array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 149 of file dla_gercond.f.

  subroutine dla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, 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, double precision, dimension( * ) RES, double precision, dimension( * ) AYB, double precision, dimension( * ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)

DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

     DLA_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 DGERFSX 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 resonsible 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by DGETRF.

    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 DGETRF; 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 DOUBLE PRECISION array, dimension (LDB,NRHS)
         The right-hand-side matrix B.

    LDB

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

    Y

              Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
         On entry, the solution matrix X, as computed by DGETRS.
         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 DLA_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 DOUBLE PRECISION array, dimension (N)
         Workspace to hold the intermediate residual.

    AYB

              AYB is DOUBLE PRECISION array, dimension (N)
         Workspace. This can be the same workspace passed for Y_TAIL.

    DY

              DY is DOUBLE PRECISION array, dimension (N)
         Workspace to hold the intermediate solution.

    Y_TAIL

              Y_TAIL is DOUBLE PRECISION 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 DGETRS had an illegal
                 value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 390 of file dla_gerfsx_extended.f.

  double precision function dla_gerpvgrw (integer N, integer NCOLS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF)

DLA_GERPVGRW

Purpose:

     DLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by DGETRF.

    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 dla_gerpvgrw.f.

  subroutine dlaorhr_col_getrfnp (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, integer INFO)

DLAORHR_COL_GETRFNP

Purpose:

     DLAORHR_COL_GETRFNP computes the modified LU factorization without
     pivoting of a real 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 DORHR_COL. In DORHR_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 DLAORHR_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 plus or minus one.

    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 dlaorhr_col_getrfnp.f.

  recursive subroutine dlaorhr_col_getrfnp2 (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, integer INFO)

DLAORHR_COL_GETRFNP2

Purpose:

     DLAORHR_COL_GETRFNP2 computes the modified LU factorization without
     pivoting of a real 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 DORHR_COL. In DORHR_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].
     DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
     routine DLAORHR_COL_GETRFNP, which uses blocked code calling
     Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
     is self-sufficient and can be used without DLAORHR_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 plus or minus one.

    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 dlaorhr_col_getrfnp2.f.

  recursive subroutine dlaqz0 (character, intent(in) WANTS, character, intent(in) WANTQ, character, intent(in) WANTZ, integer, intent(in) N, integer, intent(in) ILO, integer, intent(in) IHI, double precision, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, double precision, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, double precision, dimension( * ), intent(inout) ALPHAR, double precision, dimension( * ), intent(inout) ALPHAI, double precision, dimension( * ), intent(inout) BETA, double precision, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, double precision, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, double precision, dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, integer, intent(in) REC, integer, intent(out) INFO)

DLAQZ0

Purpose:

     DLAQZ0 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**T,  B = Q1*T*Z1**T,
     as computed by DGGHRD.
     If JOB='S', then the Hessenberg-triangular pair (H,T) is
     also reduced to generalized Schur form,
        H = Q*S*Z**T,  T = Q*P*Z**T,
     where Q and Z are orthogonal matrices, P is an upper triangular
     matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
     diagonal blocks.
     The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
     (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
     eigenvalues.
     Additionally, the 2-by-2 upper triangular diagonal blocks of P
     corresponding to 2-by-2 blocks of S are reduced to positive diagonal
     form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
     P(j,j) > 0, and P(j+1,j+1) > 0.
     Optionally, the orthogonal matrix Q from the generalized Schur
     factorization may be postmultiplied into an input matrix Q1, and the
     orthogonal matrix Z may be postmultiplied into an input matrix Z1.
     If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
     the matrix pair (A,B) to generalized upper Hessenberg form, then the
     output matrices Q1*Q and Z1*Z are the orthogonal factors from the
     generalized Schur factorization of (A,B):
        A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
     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.
     Real 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 orthogonal 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 orthogonal 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 DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the N-by-N upper Hessenberg matrix A.
              On exit, if JOB = 'S', A contains the upper quasi-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 DOUBLE PRECISION 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;
              2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
              are reduced to positive diagonal form, i.e., if A(j+1,j) is
              non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
              B(j+1,j+1) > 0.
              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 ).

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)
              The real parts of each scalar alpha defining an eigenvalue
              of GNEP.

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)
              The imaginary parts of each scalar alpha defining an
              eigenvalue of GNEP.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 DOUBLE PRECISION array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
              vectors of (A,B), and if COMPQ = 'V', the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPZ = 'I', the orthogonal matrix of
              right Schur vectors of (H,T), and if COMPZ = 'V', the
              orthogonal 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              If 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 ALPHAR(i), ALPHAI(i), and
                         BETA(i), i=INFO+1,...,N should be correct.

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 302 of file dlaqz0.f.

  subroutine dlaqz1 (double precision, dimension( lda, * ), intent(in) A, integer, intent(in) LDA, double precision, dimension( ldb, * ), intent(in) B, integer, intent(in) LDB, double precision, intent(in) SR1, double precision, intent(in) SR2, double precision, intent(in) SI, double precision, intent(in) BETA1, double precision, intent(in) BETA2, double precision, dimension( * ), intent(out) V)

DLAQZ1

Purpose:

          Given a 3-by-3 matrix pencil (A,B), DLAQZ1 sets v to a
          scalar multiple of the first column of the product
          (*)  K = (A - (beta2*sr2 - i*si)*B)*B^(-1)*(beta1*A - (sr2 + i*si2)*B)*B^(-1).
          It is assumed that either
                  1) sr1 = sr2
              or
                  2) si = 0.
          This is useful for starting double implicit shift bulges
          in the QZ algorithm.

Parameters
    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
                  The 3-by-3 matrix A in (*).

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
                  The 3-by-3 matrix B in (*).

    LDB

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

    SR1

              SR1 is DOUBLE PRECISION

    SR2

              SR2 is DOUBLE PRECISION

    SI

              SI is DOUBLE PRECISION

    BETA1

              BETA1 is DOUBLE PRECISION

    BETA2

              BETA2 is DOUBLE PRECISION

    V

              V is DOUBLE PRECISION array, dimension (N)
                  A scalar multiple of the first column of the
                  matrix K in (*).

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 125 of file dlaqz1.f.

  subroutine dlaqz2 (logical, intent(in) ILQ, logical, intent(in) ILZ, integer, intent(in) K, integer, intent(in) ISTARTM, integer, intent(in) ISTOPM, integer, intent(in) IHI, double precision, dimension( lda, * ) A, integer, intent(in) LDA, double precision, dimension( ldb, * ) B, integer, intent(in) LDB, integer, intent(in) NQ, integer, intent(in) QSTART, double precision, dimension( ldq, * ) Q, integer, intent(in) LDQ, integer, intent(in) NZ, integer, intent(in) ZSTART, double precision, dimension( ldz, * ) Z, integer, intent(in) LDZ)

DLAQZ2

Purpose:

          DLAQZ2 chases a 2x2 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+2,k:k+1),B(k+1:k+2,k:k+1)).
                  On exit, the bulge is located in
                  (A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2)).

    ISTARTM

              ISTARTM is INTEGER

    ISTOPM

              ISTOPM is INTEGER
                  Updates to (A,B) are restricted to
                  (istartm:k+3,k:istopm). It is assumed
                  without checking that istartm <= k+1 and
                  k+2 <= istopm

    IHI

              IHI is INTEGER

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)

    LDA

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

    B

              B is DOUBLE PRECISION 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+4-qStart of Q.

    Q

              Q is DOUBLE PRECISION 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+3-qStart of Z.

    Z

              Z is DOUBLE PRECISION 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 172 of file dlaqz2.f.

  recursive subroutine dlaqz3 (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, double precision, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, double precision, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, double precision, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, double precision, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, integer, intent(out) NS, integer, intent(out) ND, double precision, dimension( * ), intent(inout) ALPHAR, double precision, dimension( * ), intent(inout) ALPHAI, double precision, dimension( * ), intent(inout) BETA, double precision, dimension( ldqc, * ) QC, integer, intent(in) LDQC, double precision, dimension( ldzc, * ) ZC, integer, intent(in) LDZC, double precision, dimension( * ) WORK, integer, intent(in) LWORK, integer, intent(in) REC, integer, intent(out) INFO)

DLAQZ3

Purpose:

     DLAQZ3 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 DOUBLE PRECISION array, dimension (LDA, N)

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)

    LDB

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

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)

    LDQ

              LDQ is INTEGER

    Z

              Z is DOUBLE PRECISION 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.

    ALPHAR

              ALPHAR is DOUBLE PRECISION array, dimension (N)
              The real parts of each scalar alpha defining an eigenvalue
              of GNEP.

    ALPHAI

              ALPHAI is DOUBLE PRECISION array, dimension (N)
              The imaginary parts of each scalar alpha defining an
              eigenvalue of GNEP.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 DOUBLE PRECISION array, dimension (LDQC, NW)

    LDQC

              LDQC is INTEGER

    ZC

              ZC is DOUBLE PRECISION array, dimension (LDZC, NW)

    LDZC

              LDZ is INTEGER

    WORK

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

    LWORK

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

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 235 of file dlaqz3.f.

  subroutine dlaqz4 (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, double precision, dimension( * ), intent(inout) SR, double precision, dimension( * ), intent(inout) SI, double precision, dimension( * ), intent(inout) SS, double precision, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, double precision, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, double precision, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, double precision, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, double precision, dimension( ldqc, * ), intent(inout) QC, integer, intent(in) LDQC, double precision, dimension( ldzc, * ), intent(inout) ZC, integer, intent(in) LDZC, double precision, dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, integer, intent(out) INFO)

DLAQZ4

Purpose:

     DLAQZ4 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

    SR

              SR is DOUBLE PRECISION array. SR contains
              the real parts of the shifts to use.

    SI

              SI is DOUBLE PRECISION array. SI contains
              the imaginary parts of the shifts to use.

    SS

              SS is DOUBLE PRECISION array. SS contains
              the scale of the shifts to use.

    A

              A is DOUBLE PRECISION array, dimension (LDA, N)

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB, N)

    LDB

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

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)

    LDQ

              LDQ is INTEGER

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, N)

    LDZ

              LDZ is INTEGER

    QC

              QC is DOUBLE PRECISION array, dimension (LDQC, NBLOCK_DESIRED)

    LDQC

              LDQC is INTEGER

    ZC

              ZC is DOUBLE PRECISION array, dimension (LDZC, NBLOCK_DESIRED)

    LDZC

              LDZ is INTEGER

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              If 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 209 of file dlaqz4.f.

  subroutine dtgevc (character SIDE, character HOWMNY, logical, dimension( * ) SELECT, integer N, double precision, dimension( lds, * ) S, integer LDS, double precision, dimension( ldp, * ) P, integer LDP, double precision, dimension( ldvl, * ) VL, integer LDVL, double precision, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, double precision, dimension( * ) WORK, integer INFO)

DTGEVC

Purpose:

     DTGEVC computes some or all of the right and/or left eigenvectors of
     a pair of real matrices (S,P), where S is a quasi-triangular matrix
     and P is upper triangular.  Matrix pairs of this type are produced by
     the generalized Schur factorization of a matrix pair (A,B):
        A = Q*S*Z**T,  B = Q*P*Z**T
     as computed by DGGHRD + DHGEQZ.
     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 blocks 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 orthogonal 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.  If w(j) is a real eigenvalue, the corresponding
              real eigenvector is computed if SELECT(j) is .TRUE..
              If w(j) and w(j+1) are the real and imaginary parts of a
              complex eigenvalue, the corresponding complex eigenvector
              is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
              and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
              set to .FALSE..
              Not referenced if HOWMNY = 'A' or 'B'.

    N

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

    S

              S is DOUBLE PRECISION array, dimension (LDS,N)
              The upper quasi-triangular matrix S from a generalized Schur
              factorization, as computed by DHGEQZ.

    LDS

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

    P

              P is DOUBLE PRECISION array, dimension (LDP,N)
              The upper triangular matrix P from a generalized Schur
              factorization, as computed by DHGEQZ.
              2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
              of S must be in positive diagonal form.

    LDP

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

    VL

              VL is DOUBLE PRECISION array, dimension (LDVL,MM)
              On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
              contain an N-by-N matrix Q (usually the orthogonal matrix Q
              of left Schur vectors returned by DHGEQZ).
              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.
              A complex eigenvector corresponding to a complex eigenvalue
              is stored in two consecutive columns, the first holding the
              real part, and the second the imaginary part.
              Not referenced if SIDE = 'R'.

    LDVL

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

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,MM)
              On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
              contain an N-by-N matrix Z (usually the orthogonal matrix Z
              of right Schur vectors returned by DHGEQZ).
              On exit, if SIDE = 'R' or 'B', VR contains:
              if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
              if HOWMNY = 'B' or 'b', the matrix Z*X;
              if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                          specified by SELECT, stored consecutively in the
                          columns of VR, in the same order as their
                          eigenvalues.
              A complex eigenvector corresponding to a complex eigenvalue
              is stored in two consecutive columns, the first holding the
              real part and the second the imaginary part.
              Not referenced if SIDE = 'L'.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR.  LDVR >= 1, and if
              SIDE = 'R' or 'B', LDVR >= N.

    MM

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

    M

              M is INTEGER
              The number of columns in the arrays VL and/or VR actually
              used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
              is set to N.  Each selected real eigenvector occupies one
              column and each selected complex eigenvector occupies two
              columns.

    WORK

              WORK is DOUBLE PRECISION array, dimension (6*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
                    eigenvalue.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Allocation of workspace:
      ---------- -- ---------
         WORK( j ) = 1-norm of j-th column of A, above the diagonal
         WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
         WORK( 2*N+1:3*N ) = real part of eigenvector
         WORK( 3*N+1:4*N ) = imaginary part of eigenvector
         WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
         WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
      Rowwise vs. columnwise solution methods:
      ------- --  ---------- -------- -------
      Finding a generalized eigenvector consists basically of solving the
      singular triangular system
       (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)
      Consider finding the i-th right eigenvector (assume all eigenvalues
      are real). The equation to be solved is:
           n                   i
      0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
          k=j                 k=j
      where  C = (A - w B)  (The components v(i+1:n) are 0.)
      The "rowwise" method is:
      (1)  v(i) := 1
      for j = i-1,. . .,1:
                              i
          (2) compute  s = - sum C(j,k) v(k)   and
                            k=j+1
          (3) v(j) := s / C(j,j)
      Step 2 is sometimes called the "dot product" step, since it is an
      inner product between the j-th row and the portion of the eigenvector
      that has been computed so far.
      The "columnwise" method consists basically in doing the sums
      for all the rows in parallel.  As each v(j) is computed, the
      contribution of v(j) times the j-th column of C is added to the
      partial sums.  Since FORTRAN arrays are stored columnwise, this has
      the advantage that at each step, the elements of C that are accessed
      are adjacent to one another, whereas with the rowwise method, the
      elements accessed at a step are spaced LDS (and LDP) words apart.
      When finding left eigenvectors, the matrix in question is the
      transpose of the one in storage, so the rowwise method then
      actually accesses columns of A and B at each step, and so is the
      preferred method.

Definition at line 293 of file dtgevc.f.

  subroutine dtgexc (logical WANTQ, logical WANTZ, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ, integer IFST, integer ILST, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DTGEXC

Purpose:

     DTGEXC reorders the generalized real Schur decomposition of a real
     matrix pair (A,B) using an orthogonal equivalence transformation
                    (A, B) = Q * (A, B) * Z**T,
     so that the diagonal block of (A, B) with row index IFST is moved
     to row ILST.
     (A, B) must be in generalized real Schur canonical form (as returned
     by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
     diagonal blocks. B is upper triangular.
     Optionally, the matrices Q and Z of generalized Schur vectors are
     updated.
            Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
            Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T

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 DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the matrix A in generalized real Schur canonical
              form.
              On exit, the updated matrix A, again in generalized
              real Schur canonical form.

    LDA

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

    B

              B is DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the matrix B in generalized real Schur canonical
              form (A,B).
              On exit, the updated matrix B, again in generalized
              real Schur canonical form (A,B).

    LDB

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

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              On entry, if WANTQ = .TRUE., the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ,N)
              On entry, if WANTZ = .TRUE., the orthogonal 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.
              On exit, if IFST pointed on entry to the second row of
              a 2-by-2 block, it is changed to point to the first row;
              ILST always points to the first row of the block in its
              final position (which may differ from its input value by
              +1 or -1). 1 <= IFST, ILST <= N.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    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.

Definition at line 218 of file dtgexc.f.

  subroutine sgelqt (integer M, integer N, integer MB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO)

SGELQT

Purpose:

     DGELQT computes a blocked LQ factorization of a real 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.

    MB

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

    A

              A is REAL 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
              are the rows of V.

    LDA

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

    T

              T is REAL 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 >= MB.

    WORK

              WORK is REAL array, dimension (MB*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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (         1 v3 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/MB), where each
      block is of order MB except for the last block, which is of order
      IB = K - (B-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-K matrix T as
                   T = (T1 T2 ... TB).

Definition at line 123 of file sgelqt.f.

  recursive subroutine sgelqt3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)

SGELQT3

Purpose:

     SGELQT3 recursively computes a LQ factorization of a real 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 REAL array, dimension (LDA,N)
              On entry, the real M-by-N matrix A.  On exit, the elements on and
              below the diagonal contain the N-by-N lower triangular matrix L; the
              elements above the diagonal are the rows 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 REAL 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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (     1  v3 v3 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**T
      where V**T is the transpose of V.
      For details of the algorithm, see Elmroth and Gustavson (cited above).

Definition at line 115 of file sgelqt3.f.

  subroutine sgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer MB, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK, integer INFO)

SGEMLQT

Purpose:

     DGEMLQT overwrites the general real M-by-N matrix C with
                     SIDE = 'L'     SIDE = 'R'
     TRANS = 'N':      Q C            C Q
     TRANS = 'T':   Q**T C            C Q**T
     where Q is a real orthogonal matrix defined as the product of K
     elementary reflectors:
           Q = H(1) H(2) . . . H(K) = I - V T V**T
     generated using the compact WY representation as returned by SGELQT.
     Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

Parameters
    SIDE

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

    TRANS

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

    M

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

    N

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

    K

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

    MB

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

    V

              V is REAL array, dimension
                                   (LDV,M) if SIDE = 'L',
                                   (LDV,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              SGELQT in the first K rows of its array argument A.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,K).

    T

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

    LDT

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

    C

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

    LDC

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

    WORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 151 of file sgemlqt.f.

  recursive subroutine slaqz0 (character, intent(in) WANTS, character, intent(in) WANTQ, character, intent(in) WANTZ, integer, intent(in) N, integer, intent(in) ILO, integer, intent(in) IHI, real, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, real, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, real, dimension( * ), intent(inout) ALPHAR, real, dimension( * ), intent(inout) ALPHAI, real, dimension( * ), intent(inout) BETA, real, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, real, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, real, dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, integer, intent(in) REC, integer, intent(out) INFO)

SLAQZ0

Purpose:

     SLAQZ0 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**T,  B = Q1*T*Z1**T,
     as computed by SGGHRD.
     If JOB='S', then the Hessenberg-triangular pair (H,T) is
     also reduced to generalized Schur form,
        H = Q*S*Z**T,  T = Q*P*Z**T,
     where Q and Z are orthogonal matrices, P is an upper triangular
     matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
     diagonal blocks.
     The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
     (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
     eigenvalues.
     Additionally, the 2-by-2 upper triangular diagonal blocks of P
     corresponding to 2-by-2 blocks of S are reduced to positive diagonal
     form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
     P(j,j) > 0, and P(j+1,j+1) > 0.
     Optionally, the orthogonal matrix Q from the generalized Schur
     factorization may be postmultiplied into an input matrix Q1, and the
     orthogonal matrix Z may be postmultiplied into an input matrix Z1.
     If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
     the matrix pair (A,B) to generalized upper Hessenberg form, then the
     output matrices Q1*Q and Z1*Z are the orthogonal factors from the
     generalized Schur factorization of (A,B):
        A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
     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.
     Real 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 orthogonal 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 orthogonal 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 REAL array, dimension (LDA, N)
              On entry, the N-by-N upper Hessenberg matrix A.
              On exit, if JOB = 'S', A contains the upper quasi-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 REAL 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;
              2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
              are reduced to positive diagonal form, i.e., if A(j+1,j) is
              non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
              B(j+1,j+1) > 0.
              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 ).

    ALPHAR

              ALPHAR is REAL array, dimension (N)
              The real parts of each scalar alpha defining an eigenvalue
              of GNEP.

    ALPHAI

              ALPHAI is REAL array, dimension (N)
              The imaginary parts of each scalar alpha defining an
              eigenvalue of GNEP.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

    BETA

              BETA is REAL array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 REAL array, dimension (LDQ, N)
              On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
              vectors of (A,B), and if COMPQ = 'V', the orthogonal 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 REAL array, dimension (LDZ, N)
              On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
              the reduction of (A,B) to generalized Hessenberg form.
              On exit, if COMPZ = 'I', the orthogonal matrix of
              right Schur vectors of (H,T), and if COMPZ = 'V', the
              orthogonal 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 REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              If 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 ALPHAR(i), ALPHAI(i), and
                         BETA(i), i=INFO+1,...,N should be correct.

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 300 of file slaqz0.f.

  subroutine slaqz1 (real, dimension( lda, * ), intent(in) A, integer, intent(in) LDA, real, dimension( ldb, * ), intent(in) B, integer, intent(in) LDB, real, intent(in) SR1, real, intent(in) SR2, real, intent(in) SI, real, intent(in) BETA1, real, intent(in) BETA2, real, dimension( * ), intent(out) V)

SLAQZ1

Purpose:

          Given a 3-by-3 matrix pencil (A,B), SLAQZ1 sets v to a
          scalar multiple of the first column of the product
          (*)  K = (A - (beta2*sr2 - i*si)*B)*B^(-1)*(beta1*A - (sr2 + i*si2)*B)*B^(-1).
          It is assumed that either
                  1) sr1 = sr2
              or
                  2) si = 0.
          This is useful for starting double implicit shift bulges
          in the QZ algorithm.

Parameters
    A

              A is REAL array, dimension (LDA,N)
                  The 3-by-3 matrix A in (*).

    LDA

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

    B

              B is REAL array, dimension (LDB,N)
                  The 3-by-3 matrix B in (*).

    LDB

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

    SR1

              SR1 is REAL

    SR2

              SR2 is REAL

    SI

              SI is REAL

    BETA1

              BETA1 is REAL

    BETA2

              BETA2 is REAL

    V

              V is REAL array, dimension (N)
                  A scalar multiple of the first column of the
                  matrix K in (*).

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 125 of file slaqz1.f.

  subroutine slaqz2 (logical, intent(in) ILQ, logical, intent(in) ILZ, integer, intent(in) K, integer, intent(in) ISTARTM, integer, intent(in) ISTOPM, integer, intent(in) IHI, real, dimension( lda, * ) A, integer, intent(in) LDA, real, dimension( ldb, * ) B, integer, intent(in) LDB, integer, intent(in) NQ, integer, intent(in) QSTART, real, dimension( ldq, * ) Q, integer, intent(in) LDQ, integer, intent(in) NZ, integer, intent(in) ZSTART, real, dimension( ldz, * ) Z, integer, intent(in) LDZ)

SLAQZ2

Purpose:

          SLAQZ2 chases a 2x2 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+2,k:k+1),B(k+1:k+2,k:k+1)).
                  On exit, the bulge is located in
                  (A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2)).

    ISTARTM

              ISTARTM is INTEGER

    ISTOPM

              ISTOPM is INTEGER
                  Updates to (A,B) are restricted to
                  (istartm:k+3,k:istopm). It is assumed
                  without checking that istartm <= k+1 and
                  k+2 <= istopm

    IHI

              IHI is INTEGER

    A

              A is REAL array, dimension (LDA,N)

    LDA

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

    B

              B is REAL 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+4-qStart of Q.

    Q

              Q is REAL 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+3-qStart of Z.

    Z

              Z is REAL 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 slaqz2.f.

  recursive subroutine slaqz3 (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, real, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, real, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, real, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, real, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, integer, intent(out) NS, integer, intent(out) ND, real, dimension( * ), intent(inout) ALPHAR, real, dimension( * ), intent(inout) ALPHAI, real, dimension( * ), intent(inout) BETA, real, dimension( ldqc, * ) QC, integer, intent(in) LDQC, real, dimension( ldzc, * ) ZC, integer, intent(in) LDZC, real, dimension( * ) WORK, integer, intent(in) LWORK, integer, intent(in) REC, integer, intent(out) INFO)

SLAQZ3

Purpose:

     SLAQZ3 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 REAL array, dimension (LDA, N)

    LDA

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

    B

              B is REAL array, dimension (LDB, N)

    LDB

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

    Q

              Q is REAL array, dimension (LDQ, N)

    LDQ

              LDQ is INTEGER

    Z

              Z is REAL 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.

    ALPHAR

              ALPHAR is REAL array, dimension (N)
              The real parts of each scalar alpha defining an eigenvalue
              of GNEP.

    ALPHAI

              ALPHAI is REAL array, dimension (N)
              The imaginary parts of each scalar alpha defining an
              eigenvalue of GNEP.
              If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
              positive, then the j-th and (j+1)-st eigenvalues are a
              complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

    BETA

              BETA is REAL array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.
              Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 REAL array, dimension (LDQC, NW)

    LDQC

              LDQC is INTEGER

    ZC

              ZC is REAL array, dimension (LDZC, NW)

    LDZC

              LDZ is INTEGER

    WORK

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

    LWORK

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

Author
    Thijs Steel, KU Leuven

Date
    May 2020

Definition at line 234 of file slaqz3.f.

  subroutine slaqz4 (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, real, dimension( * ), intent(inout) SR, real, dimension( * ), intent(inout) SI, real, dimension( * ), intent(inout) SS, real, dimension( lda, * ), intent(inout) A, integer, intent(in) LDA, real, dimension( ldb, * ), intent(inout) B, integer, intent(in) LDB, real, dimension( ldq, * ), intent(inout) Q, integer, intent(in) LDQ, real, dimension( ldz, * ), intent(inout) Z, integer, intent(in) LDZ, real, dimension( ldqc, * ), intent(inout) QC, integer, intent(in) LDQC, real, dimension( ldzc, * ), intent(inout) ZC, integer, intent(in) LDZC, real, dimension( * ), intent(inout) WORK, integer, intent(in) LWORK, integer, intent(out) INFO)

SLAQZ4

Purpose:

     SLAQZ4 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

    SR

              SR is REAL array. SR contains
              the real parts of the shifts to use.

    SI

              SI is REAL array. SI contains
              the imaginary parts of the shifts to use.

    SS

              SS is REAL array. SS contains
              the scale of the shifts to use.

    A

              A is REAL array, dimension (LDA, N)

    LDA

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

    B

              B is REAL array, dimension (LDB, N)

    LDB

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

    Q

              Q is REAL array, dimension (LDQ, N)

    LDQ

              LDQ is INTEGER

    Z

              Z is REAL array, dimension (LDZ, N)

    LDZ

              LDZ is INTEGER

    QC

              QC is REAL array, dimension (LDQC, NBLOCK_DESIRED)

    LDQC

              LDQC is INTEGER

    ZC

              ZC is REAL array, dimension (LDZC, NBLOCK_DESIRED)

    LDZC

              LDZ is INTEGER

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,N).
              If 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 210 of file slaqz4.f.

  subroutine zgelqt (integer M, integer N, integer MB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) WORK, integer INFO)

ZGELQT

Purpose:

     ZGELQT computes a blocked LQ 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.

    MB

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

    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
              are the rows 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 >= MB.

    WORK

              WORK is COMPLEX*16 array, dimension (MB*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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (         1 v3 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/MB), where each
      block is of order MB except for the last block, which is of order
      IB = K - (B-1)*MB.  For each of the B blocks, a upper triangular block
      reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
      for the last block) T's are stored in the MB-by-K matrix T as
                   T = (T1 T2 ... TB).

Definition at line 138 of file zgelqt.f.

  recursive subroutine zgelqt3 (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, integer INFO)

ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

     ZGELQT3 recursively computes a LQ 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
              below the diagonal contain the N-by-N lower triangular matrix L; the
              elements above the diagonal are the rows 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 row
      above the diagonal. For example, if M=5 and N=3, the matrix V is
                   V = (  1  v1 v1 v1 v1 )
                       (     1  v2 v2 v2 )
                       (     1  v3 v3 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**T
      where V**T is the transpose of V.
      For details of the algorithm, see Elmroth and Gustavson (cited above).

Definition at line 130 of file zgelqt3.f.

  subroutine zgemlqt (character SIDE, character TRANS, integer M, integer N, integer K, integer MB, 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)

ZGEMLQT

Purpose:

     ZGEMLQT overwrites the general real 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 ZGELQT.
     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.

    MB

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

    V

              V is COMPLEX*16 array, dimension
                                   (LDV,M) if SIDE = 'L',
                                   (LDV,N) if SIDE = 'R'
              The i-th row must contain the vector which defines the
              elementary reflector H(i), for i = 1,2,...,k, as returned by
              ZGELQT in the first K rows of its array argument A.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,K).

    T

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

    LDT

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

    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*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 166 of file zgemlqt.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.