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realGEcomputational(3) LAPACK realGEcomputational(3)

NAME

realGEcomputational

SYNOPSIS

Functions


subroutine sgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
SGEBAK subroutine sgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
SGEBAL subroutine sgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. subroutine sgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
SGEBRD subroutine sgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SGECON subroutine sgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQU subroutine sgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQUB subroutine sgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. subroutine sgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
SGEHRD subroutine sgelq2 (M, N, A, LDA, TAU, WORK, INFO)
SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm. subroutine sgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGELQF subroutine sgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO)
SGEMQRT subroutine sgeql2 (M, N, A, LDA, TAU, WORK, INFO)
SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm. subroutine sgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQLF subroutine sgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
SGEQP3 subroutine sgeqr2 (M, N, A, LDA, TAU, WORK, INFO)
SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm. subroutine sgeqr2p (M, N, A, LDA, TAU, WORK, INFO)
SGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm. subroutine sgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF subroutine sgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRFP subroutine sgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO)
SGEQRT subroutine sgeqrt2 (M, N, A, LDA, T, LDT, INFO)
SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. recursive subroutine sgeqrt3 (M, N, A, LDA, T, LDT, INFO)
SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. subroutine sgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGERFS subroutine sgerfsx (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)
SGERFSX subroutine sgerq2 (M, N, A, LDA, TAU, WORK, INFO)
SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm. subroutine sgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGERQF subroutine sgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO)
SGESVJ subroutine sgetf2 (M, N, A, LDA, IPIV, INFO)
SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm). subroutine sgetrf (M, N, A, LDA, IPIV, INFO)
SGETRF recursive subroutine sgetrf2 (M, N, A, LDA, IPIV, INFO)
SGETRF2 subroutine sgetri (N, A, LDA, IPIV, WORK, LWORK, INFO)
SGETRI subroutine sgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS subroutine shgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ subroutine sla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds. real function sla_gercond (TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
SLA_GERCOND estimates the Skeel condition number for a general matrix. subroutine sla_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)
SLA_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. real function sla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF)
SLA_GERPVGRW subroutine stgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
STGEVC subroutine stgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
STGEXC

Detailed Description

This is the group of real computational functions for GE matrices

Function Documentation

subroutine sgebak (character JOB, character SIDE, integer N, integer ILO, integer IHI, real, dimension( * ) SCALE, integer M, real, dimension( ldv, * ) V, integer LDV, integer INFO)

SGEBAK

Purpose:

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

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 SGEBAL.

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 SGEBAL.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

SCALE

          SCALE is REAL array, dimension (N)
          Details of the permutation and scaling factors, as returned
          by SGEBAL.

M

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

V

          V is REAL array, dimension (LDV,M)
          On entry, the matrix of right or left eigenvectors to be
          transformed, as returned by SHSEIN or STREVC.
          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.

Date:

December 2016

subroutine sgebal (character JOB, integer N, real, dimension( lda, * ) A, integer LDA, integer ILO, integer IHI, real, dimension( * ) SCALE, integer INFO)

SGEBAL

Purpose:

 SGEBAL 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 REAL 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 REAL 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.

Date:

December 2016

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 sgebd2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer INFO)

SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

 SGEBD2 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 REAL 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 REAL array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B:
          D(i) = A(i,i).

E

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

TAUP

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

WORK

          WORK is REAL 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.

Date:

June 2017

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 sgebrd (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEBRD

Purpose:

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

E

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

TAUP

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

WORK

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

LWORK

          LWORK is INTEGER
          The 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.

Date:

November 2017

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 sgecon (character NORM, integer N, real, dimension( lda, * ) A, integer LDA, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SGECON

Purpose:

 SGECON 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 SGETRF.
 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 REAL array, dimension (LDA,N)
          The factors L and U from the factorization A = P*L*U
          as computed by SGETRF.

LDA

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

ANORM

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

WORK

          WORK is REAL array, dimension (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.

Date:

December 2016

subroutine sgeequ (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)

SGEEQU

Purpose:

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

C

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

ROWCND

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

INFO

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

Date:

December 2016

subroutine sgeequb (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, integer INFO)

SGEEQUB

Purpose:

 SGEEQUB 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 SGEEQU 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 REAL 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 REAL array, dimension (M)
          If INFO = 0 or INFO > M, R contains the row scale factors
          for A.

C

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

ROWCND

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

INFO

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

Date:

December 2016

subroutine sgehd2 (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

 SGEHD2 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 SGEBAL; otherwise they should be
          set to 1 and N respectively. See Further Details.
          1 <= ILO <= IHI <= max(1,N).

A

          A is REAL array, dimension (LDA,N)
          On entry, the n by n general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          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 REAL array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

          WORK is REAL array, dimension (N)

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

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 sgehrd (integer N, integer ILO, integer IHI, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEHRD

Purpose:

 SGEHRD 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 SGEBAL; 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 REAL array, dimension (LDA,N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          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 REAL 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 REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The length of the array WORK.  LWORK >= 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.

Date:

December 2016

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 sgelq2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)

SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

 SGELQ2 computes an LQ factorization of a real m by n matrix A:
 A = L * 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 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,
          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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

          WORK is REAL 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.

Date:

December 2016

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 sgelqf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGELQF

Purpose:

 SGELQF computes an LQ factorization of a real M-by-N matrix A:
 A = L * 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 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,
          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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

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

LWORK

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

Date:

December 2016

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

SGEMQRT

Purpose:

 SGEMQRT 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 SGEQRT.
 Q is of order M if SIDE = 'L' and of order N  if SIDE = 'R'.

Parameters:

SIDE

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

TRANS

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

M

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

N

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

K

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

NB

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

V

          V is REAL array, dimension (LDV,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CGEQRT in the first K columns of its array argument A.

LDV

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

T

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

LDT

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

C

          C is 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*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.

Date:

December 2016

subroutine sgeql2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)

SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

 SGEQL2 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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

          WORK is REAL array, dimension (N)

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

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 sgeqlf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEQLF

Purpose:

 SGEQLF 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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

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

LWORK

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

Date:

December 2016

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 sgeqp3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEQP3

Purpose:

 SGEQP3 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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

WORK

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

LWORK

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

Date:

December 2016

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 sgeqr2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)

SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

 SGEQR2 computes a QR factorization of a real m by n matrix A:
 A = Q * R.

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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

          WORK is REAL array, dimension (N)

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

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 sgeqr2p (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)

SGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.

Purpose:

 SGEQR2P computes a QR factorization of a real m by n matrix A:
 A = Q * R. The diagonal entries of R are nonnegative.

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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

          WORK is REAL array, dimension (N)

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

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 sgeqrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEQRF

Purpose:

 SGEQRF computes a QR factorization of a real M-by-N matrix A:
 A = Q * R.

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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

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

LWORK

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

Date:

December 2016

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 sgeqrfp (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEQRFP

Purpose:

 SGEQRFP computes a QR factorization of a real M-by-N matrix A:
 A = Q * R. The diagonal entries of R are nonnegative.

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 REAL 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

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

LWORK

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

Date:

December 2016

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 sgeqrt (integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO)

SGEQRT

Purpose:

 SGEQRT 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 REAL 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 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 >= NB.

WORK

          WORK is REAL array, dimension (NB*N)

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

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 sgeqrt2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)

SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

 SGEQRT2 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 REAL 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 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.

Date:

December 2016

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 sgeqrt3 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO)

SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:

 SGEQRT3 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 REAL 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 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.

Date:

June 2016

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

SGERFS

Purpose:

 SGERFS 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 REAL 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 REAL array, dimension (LDAF,N)
          The factors L and U from the factorization A = P*L*U
          as computed by SGETRF.

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

B

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

LDB

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

X

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

LDX

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

FERR

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

BERR

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

WORK

          WORK is REAL array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

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

Internal Parameters:

  ITMAX is the maximum number of steps of iterative refinement.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine sgerfsx (character TRANS, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SGERFSX

Purpose:

    SGERFSX 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 REAL 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 REAL array, dimension (LDAF,N)
     The factors L and U from the factorization A = P*L*U
     as computed by SGETRF.

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

R

          R is REAL 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 REAL 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 REAL array, dimension (LDB,NRHS)
     The right hand side matrix B.

LDB

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

X

          X is REAL array, dimension (LDX,NRHS)
     On entry, the solution matrix X, as computed by SGETRS.
     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 REAL
     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 REAL 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 REAL 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) * 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.
     See Lapack Working Note 165 for further details and extra
     cautions.

ERR_BNDS_COMP

          ERR_BNDS_COMP is REAL 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 .LT. 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) * 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.
     See Lapack Working Note 165 for further details and extra
     cautions.

NPARAMS

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

PARAMS

          PARAMS is REAL array, dimension NPARAMS
     Specifies algorithm parameters.  If an entry is .LT. 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: 1.0
            = 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 REAL 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.

Date:

December 2016

subroutine sgerq2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO)

SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

Purpose:

 SGERQ2 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 REAL array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, if m <= n, the upper triangle of the subarray
          A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
          if m >= n, the elements on and above the (m-n)-th subdiagonal
          contain the m by n upper trapezoidal matrix R; the remaining
          elements, with the array 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

          WORK is REAL 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.

Date:

December 2016

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 sgerqf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGERQF

Purpose:

 SGERQF 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 REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if m <= n, the upper triangle of the subarray
          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
          if m >= n, the elements on and above the (m-n)-th subdiagonal
          contain the M-by-N upper trapezoidal matrix R;
          the remaining elements, with the array 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 REAL array, dimension (min(M,N))
          The scalar factors of the elementary reflectors (see Further
          Details).

WORK

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

LWORK

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

Date:

December 2016

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

SGESVJ

Purpose:

 SGESVJ 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.
 SGESVJ 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=SQRT(M), EPS=SLAMCH('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/SLAMCH('E') > M >= 0.

N

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

A

          A is REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C':
                 If INFO .EQ. 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 SLAMCH('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=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
                 see the description of JOBU.
                 If INFO .GT. 0,
                 the procedure SGESVJ 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 .EQ. 'N':
                 If INFO .EQ. 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 .GT. 0 :
                 the procedure SGESVJ 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 REAL array, dimension (N)
          On exit,
          If INFO .EQ. 0 :
          depending on the value SCALE = WORK(1), we have:
                 If SCALE .EQ. 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 .GT. 0 :
          the procedure SGESVJ 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 .EQ. 'A', then the product of Jacobi rotations in SGESVJ
          is applied to the first MV rows of V. See the description of JOBV.

V

          V is REAL 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 .GE. 1.
          If JOBV .EQ. 'V', then LDV .GE. max(1,N).
          If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .

WORK

          WORK is REAL array, dimension (LWORK)
          On entry,
          If JOBU .EQ. '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=SLAMCH('E').
                    It is required that CTOL >= ONE, i.e. it is not
                    allowed to force the routine to obtain orthogonality
                    below EPSILON.
          On exit,
          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
                    are the computed singular vcalues 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 SGESVJ did
                    not converge, as it can be used to estimate whether
                    the output is stil 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 : SGESVJ 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.

Date:

June 2017

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 tunning parameters (marked with [TP]) are available for the implementer. The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits.

Contributors:

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

References:

[1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling. SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.

[2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer. SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.

[3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR. [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic. SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.

[5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. LAPACK Working note 169.

[6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. LAPACK Working note 170.

[7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations. Department of Mathematics, University of Zagreb, 2008.

Bugs, Examples and Comments:

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

subroutine sgetf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).

Purpose:

 SGETF2 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 REAL 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.

Date:

December 2016

subroutine sgetrf (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

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

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

Purpose:

 SGETRF 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 REAL 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:

 SGETRF 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 REAL 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:

 SGETRF 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 STRSM, 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 REAL 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 sgetrf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

SGETRF2

Purpose:

 SGETRF2 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 REAL 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:

June 2016

subroutine sgetri (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGETRI

Purpose:

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

WORK

          WORK is REAL 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.

Date:

December 2016

subroutine sgetrs (character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SGETRS

Purpose:

 SGETRS 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 SGETRF.

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

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

B

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

LDB

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

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

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

SHGEQZ

Purpose:

 SHGEQZ 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.

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 REAL 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 REAL 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 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 (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 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.

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.

Date:

June 2016

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 sla_geamv (integer TRANS, integer M, integer N, real ALPHA, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)

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

Purpose:

 SLA_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 REAL
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.

A

          A is REAL 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 REAL 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 REAL
           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 REAL 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.

Date:

June 2017

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

SLA_GERCOND estimates the Skeel condition number for a general matrix.

Purpose:

    SLA_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 REAL 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 REAL array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by SGETRF.

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 SGETRF; 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 REAL 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 REAL array, dimension (3*N).
     Workspace.

IWORK

          IWORK is INTEGER array, dimension (N).
     Workspace.2

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine sla_gerfsx_extended (integer PREC_TYPE, integer TRANS_TYPE, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERRS_N, real, dimension( nrhs, * ) ERRS_C, real, dimension( * ) RES, real, dimension( * ) AYB, real, dimension( * ) DY, real, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)

SLA_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:

 SLA_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 SGERFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERRS_N
 and ERRS_C for details of the error bounds. Note that this
 subroutine is only resonsible for setting the second fields of
 ERRS_N and ERRS_C.

Parameters:

PREC_TYPE

          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and
     P    = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X', '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 REAL 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 REAL array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by SGETRF.

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 SGETRF; 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 REAL 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 REAL array, dimension (LDB,NRHS)
     The right-hand-side matrix B.

LDB

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

Y

          Y is REAL array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by SGETRS.
     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 REAL 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 SLA_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 REAL 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 REAL 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 .LT. 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 REAL array, dimension (N)
     Workspace to hold the intermediate residual.

AYB

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

DY

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

Y_TAIL

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

RCOND

          RCOND is REAL
     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 REAL
     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 REAL
     Determines when to start considering componentwise convergence.
     Componentwise convergence is only considered after each component
     of the solution Y is stable, which we definte 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 SGETRS had an illegal
             value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

real function sla_gerpvgrw (integer N, integer NCOLS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF)

SLA_GERPVGRW

Purpose:

 SLA_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 REAL 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 REAL array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by SGETRF.

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.

Date:

December 2016

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

STGEVC

Purpose:

 STGEVC 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 SGGHRD + SHGEQZ.
 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 REAL array, dimension (LDS,N)
          The upper quasi-triangular matrix S from a generalized Schur
          factorization, as computed by SHGEQZ.

LDS

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

P

          P is REAL array, dimension (LDP,N)
          The upper triangular matrix P from a generalized Schur
          factorization, as computed by SHGEQZ.
          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 REAL array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of left Schur vectors returned by SHGEQZ).
          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 REAL 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 SHGEQZ).
          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 REAL 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.

Date:

December 2016

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

STGEXC

Purpose:

 STGEXC 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 SGGES), 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 REAL 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 REAL 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 REAL 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 REAL 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 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 >= 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.

Date:

June 2017

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.

Author

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