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).

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).

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 complex M-by-N matrix C with
SIDE = 'L' SIDE = 'R' TRANS = 'N': Q C C Q TRANS = 'C': Q**H C C Q**H
where Q is a complex unitary 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.

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.

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

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).

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).

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.

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.

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.

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).

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.)

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).

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).

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).

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).

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.

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 DGEQRT.
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
          DGEQRT 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 DGEQRT, 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.

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).

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).

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

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).

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

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 >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise.
          For optimum performance LWORK >= N*NB, where NB is
          the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**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).

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

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).

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.

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).

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.

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.

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).

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 >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
 
  The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(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).

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
  [email protected]. Thank you.

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.

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

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.

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.

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.

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.

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.

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.

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 responsible for setting the second fields of
 ERRS_N and ERRS_C.
Parameters
PREC_TYPE
          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and P
          = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X' or 'E':  Extra
TRANS_TYPE
          TRANS_TYPE is INTEGER
     Specifies the transposition operation on A.
     The value is defined by ILATRANS(T) where T is a CHARACTER and T
          = 'N':  No transpose
          = 'T':  Transpose
          = 'C':  Conjugate transpose
N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
NRHS
          NRHS is INTEGER
     The number of right-hand-sides, i.e., the number of columns of the
     matrix B.
A
          A is 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.

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.

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

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

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

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

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

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

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

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.

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.

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).

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).

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.

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

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

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

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

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

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).

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).

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 complex M-by-N matrix C with
SIDE = 'L' SIDE = 'R' TRANS = 'N': Q C C Q TRANS = 'C': Q**H C C Q**H
where Q is a complex unitary 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.

Author

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