realGEsolve(3)

NAME
realGEsolve

SYNOPSIS

  Functions
subroutine sgels (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO)
SGELS solves overdetermined or underdetermined systems for GE matrices
subroutine sgelsd (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO)
SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices
subroutine sgelss (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO)
SGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine sgelsy (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO)
SGELSY solves overdetermined or underdetermined systems for GE matrices
subroutine sgesv (N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGESV computes the solution to system of linear equations A * X = B for GE matrices (simple driver)
subroutine sgesvx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SGESVX computes the solution to system of linear equations A * X = B for GE matrices
subroutine sgesvxx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
SGESVXX computes the solution to system of linear equations A * X = B for GE matrices

Detailed Description

This is the group of real solve driver functions for GE matrices

Function Documentation

  subroutine sgels (character TRANS, integer M, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

     SGELS solves overdetermined or underdetermined real linear systems
     involving an M-by-N matrix A, or its transpose, using a QR or LQ
     factorization of A.  It is assumed that A has full rank.
     The following options are provided:
     1. If TRANS = 'N' and m >= n:  find the least squares solution of
        an overdetermined system, i.e., solve the least squares problem
                     minimize || B - A*X ||.
     2. If TRANS = 'N' and m < n:  find the minimum norm solution of
        an underdetermined system A * X = B.
     3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
        an underdetermined system A**T * X = B.
     4. If TRANS = 'T' and m < n:  find the least squares solution of
        an overdetermined system, i.e., solve the least squares problem
                     minimize || B - A**T * X ||.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.

Parameters:
    TRANS

              TRANS is CHARACTER*1
              = 'N': the linear system involves A;
              = 'T': the linear system involves A**T.

    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.

    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)
              On entry, the M-by-N matrix A.
              On exit,
                if M >= N, A is overwritten by details of its QR
                           factorization as returned by SGEQRF;
                if M <  N, A is overwritten by details of its LQ
                           factorization as returned by SGELQF.

    LDA

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

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the matrix B of right hand side vectors, stored
              columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
              if TRANS = 'T'.
              On exit, if INFO = 0, B is overwritten by the solution
              vectors, stored columnwise:
              if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
              squares solution vectors; the residual sum of squares for the
              solution in each column is given by the sum of squares of
              elements N+1 to M in that column;
              if TRANS = 'N' and m < n, rows 1 to N of B contain the
              minimum norm solution vectors;
              if TRANS = 'T' and m >= n, rows 1 to M of B contain the
              minimum norm solution vectors;
              if TRANS = 'T' and m < n, rows 1 to M of B contain the
              least squares solution vectors; the residual sum of squares
              for the solution in each column is given by the sum of
              squares of elements M+1 to N in that column.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= MAX(1,M,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, MN + max( MN, NRHS ) ).
              For optimal performance,
              LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
              where MN = min(M,N) and NB is the optimum block size.
              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, the i-th diagonal element of the
                    triangular factor of A is zero, so that A does not have
                    full rank; the least squares solution could not be
                    computed.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 185 of file sgels.f.

  subroutine sgelsd (integer M, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) S, real RCOND, integer RANK, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Purpose:

     SGELSD computes the minimum-norm solution to a real linear least
     squares problem:
         minimize 2-norm(| b - A*x |)
     using the singular value decomposition (SVD) of A. A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.
     The problem is solved in three steps:
     (1) Reduce the coefficient matrix A to bidiagonal form with
         Householder transformations, reducing the original problem
         into a "bidiagonal least squares problem" (BLS)
     (2) Solve the BLS using a divide and conquer approach.
     (3) Apply back all the Householder transformations to solve
         the original least squares problem.
     The effective rank of A is determined by treating as zero those
     singular values which are less than RCOND times the largest singular
     value.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters:
    M

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

    N

              N is INTEGER
              The number of columns of 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)
              On entry, the M-by-N matrix A.
              On exit, A has been destroyed.

    LDA

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

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution
              matrix X.  If m >= n and RANK = n, the residual
              sum-of-squares for the solution in the i-th column is given
              by the sum of squares of elements n+1:m in that column.

    LDB

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

    S

              S is REAL array, dimension (min(M,N))
              The singular values of A in decreasing order.
              The condition number of A in the 2-norm = S(1)/S(min(m,n)).

    RCOND

              RCOND is REAL
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as zero.
              If RCOND < 0, machine precision is used instead.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the number of singular values
              which are greater than RCOND*S(1).

    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 must be at least 1.
              The exact minimum amount of workspace needed depends on M,
              N and NRHS. As long as LWORK is at least
                  12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
              if M is greater than or equal to N or
                  12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
              if M is less than N, the code will execute correctly.
              SMLSIZ is returned by ILAENV and is equal to the maximum
              size of the subproblems at the bottom of the computation
              tree (usually about 25), and
                 NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
              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 array WORK and the
              minimum size of the array IWORK, and returns these values as
              the first entries of the WORK and IWORK arrays, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
              where MINMN = MIN( M,N ).
              On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  the algorithm for computing the SVD failed to converge;
                    if INFO = i, i off-diagonal elements of an intermediate
                    bidiagonal form did not converge to zero.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    June 2017

Contributors:
    Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
     Osni Marques, LBNL/NERSC, USA

Definition at line 212 of file sgelsd.f.

  subroutine sgelss (integer M, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) S, real RCOND, integer RANK, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGELSS solves overdetermined or underdetermined systems for GE matrices

Purpose:

     SGELSS computes the minimum norm solution to a real linear least
     squares problem:
     Minimize 2-norm(| b - A*x |).
     using the singular value decomposition (SVD) of A. A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
     X.
     The effective rank of A is determined by treating as zero those
     singular values which are less than RCOND times the largest singular
     value.

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.

    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)
              On entry, the M-by-N matrix A.
              On exit, the first min(m,n) rows of A are overwritten with
              its right singular vectors, stored rowwise.

    LDA

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

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the M-by-NRHS right hand side matrix B.
              On exit, B is overwritten by the N-by-NRHS solution
              matrix X.  If m >= n and RANK = n, the residual
              sum-of-squares for the solution in the i-th column is given
              by the sum of squares of elements n+1:m in that column.

    LDB

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

    S

              S is REAL array, dimension (min(M,N))
              The singular values of A in decreasing order.
              The condition number of A in the 2-norm = S(1)/S(min(m,n)).

    RCOND

              RCOND is REAL
              RCOND is used to determine the effective rank of A.
              Singular values S(i) <= RCOND*S(1) are treated as zero.
              If RCOND < 0, machine precision is used instead.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the number of singular values
              which are greater than RCOND*S(1).

    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, and also:
              LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
              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.
              > 0:  the algorithm for computing the SVD failed to converge;
                    if INFO = i, i off-diagonal elements of an intermediate
                    bidiagonal form did not converge to zero.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 174 of file sgelss.f.

  subroutine sgelsy (integer M, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer, dimension( * ) JPVT, real RCOND, integer RANK, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGELSY solves overdetermined or underdetermined systems for GE matrices

Purpose:

     SGELSY computes the minimum-norm solution to a real linear least
     squares problem:
         minimize || A * X - B ||
     using a complete orthogonal factorization of A.  A is an M-by-N
     matrix which may be rank-deficient.
     Several right hand side vectors b and solution vectors x can be
     handled in a single call; they are stored as the columns of the
     M-by-NRHS right hand side matrix B and the N-by-NRHS solution
     matrix X.
     The routine first computes a QR factorization with column pivoting:
         A * P = Q * [ R11 R12 ]
                     [  0  R22 ]
     with R11 defined as the largest leading submatrix whose estimated
     condition number is less than 1/RCOND.  The order of R11, RANK,
     is the effective rank of A.
     Then, R22 is considered to be negligible, and R12 is annihilated
     by orthogonal transformations from the right, arriving at the
     complete orthogonal factorization:
        A * P = Q * [ T11 0 ] * Z
                    [  0  0 ]
     The minimum-norm solution is then
        X = P * Z**T [ inv(T11)*Q1**T*B ]
                     [        0         ]
     where Q1 consists of the first RANK columns of Q.
     This routine is basically identical to the original xGELSX except
     three differences:
       o The call to the subroutine xGEQPF has been substituted by the
         the call to the subroutine xGEQP3. This subroutine is a Blas-3
         version of the QR factorization with column pivoting.
       o Matrix B (the right hand side) is updated with Blas-3.
       o The permutation of matrix B (the right hand side) is faster and
         more simple.

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.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of
              columns of matrices B and X. NRHS >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A has been overwritten by details of its
              complete orthogonal factorization.

    LDA

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

    B

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

    LDB

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

    JPVT

              JPVT is INTEGER array, dimension (N)
              On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
              to the front of AP, otherwise column i is a free column.
              On exit, if JPVT(i) = k, then the i-th column of AP
              was the k-th column of A.

    RCOND

              RCOND is REAL
              RCOND is used to determine the effective rank of A, which
              is defined as the order of the largest leading triangular
              submatrix R11 in the QR factorization with pivoting of A,
              whose estimated condition number < 1/RCOND.

    RANK

              RANK is INTEGER
              The effective rank of A, i.e., the order of the submatrix
              R11.  This is the same as the order of the submatrix T11
              in the complete orthogonal factorization of A.

    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.
              The unblocked strategy requires that:
                 LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
              where MN = min( M, N ).
              The block algorithm requires that:
                 LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
              where NB is an upper bound on the blocksize returned
              by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR,
              and SORMRZ.
              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

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
     E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
     G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

Definition at line 206 of file sgelsy.f.

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

SGESV computes the solution to system of linear equations A * X = B for GE matrices (simple driver)

Purpose:

     SGESV computes the solution to a real system of linear equations
        A * X = B,
     where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
     The LU decomposition with partial pivoting and row interchanges is
     used to factor A as
        A = P * L * U,
     where P is a permutation matrix, L is unit lower triangular, and U is
     upper triangular.  The factored form of A is then used to solve the
     system of equations A * X = B.

Parameters:
    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.  NRHS >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the N-by-N coefficient matrix A.
              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,N).

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices that define the permutation matrix P;
              row i of the matrix was interchanged with row IPIV(i).

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the N-by-NRHS matrix of right hand side matrix B.
              On exit, if INFO = 0, the N-by-NRHS solution matrix X.

    LDB

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
                    has been completed, but the factor U is exactly
                    singular, so the solution could not be computed.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 124 of file sgesv.f.

  subroutine sgesvx (character FACT, character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SGESVX computes the solution to system of linear equations A * X = B for GE matrices

Purpose:

     SGESVX uses the LU factorization to compute the solution to a real
     system of linear equations
        A * X = B,
     where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
     Error bounds on the solution and a condition estimate are also
     provided.

Description:

     The following steps are performed:
     1. If FACT = 'E', real scaling factors are computed to equilibrate
        the system:
           TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
           TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
           TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
        Whether or not the system will be equilibrated depends on the
        scaling of the matrix A, but if equilibration is used, A is
        overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
        or diag(C)*B (if TRANS = 'T' or 'C').
     2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
        matrix A (after equilibration if FACT = 'E') as
           A = P * L * U,
        where P is a permutation matrix, L is a unit lower triangular
        matrix, and U is upper triangular.
     3. If some U(i,i)=0, so that U is exactly singular, then the routine
        returns with INFO = i. Otherwise, the factored form of A is used
        to estimate the condition number of the matrix A.  If the
        reciprocal of the condition number is less than machine precision,
        INFO = N+1 is returned as a warning, but the routine still goes on
        to solve for X and compute error bounds as described below.
     4. The system of equations is solved for X using the factored form
        of A.
     5. Iterative refinement is applied to improve the computed solution
        matrix and calculate error bounds and backward error estimates
        for it.
     6. If equilibration was used, the matrix X is premultiplied by
        diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
        that it solves the original system before equilibration.

Parameters:
    FACT

              FACT is CHARACTER*1
              Specifies whether or not the factored form of the matrix A is
              supplied on entry, and if not, whether the matrix A should be
              equilibrated before it is factored.
              = 'F':  On entry, AF and IPIV contain the factored form of A.
                      If EQUED is not 'N', the matrix A has been
                      equilibrated with scaling factors given by R and C.
                      A, AF, and IPIV are not modified.
              = 'N':  The matrix A will be copied to AF and factored.
              = 'E':  The matrix A will be equilibrated if necessary, then
                      copied to AF and factored.

    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  (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 matrices B and X.  NRHS >= 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
              not 'N', then A must have been equilibrated by the scaling
              factors in R and/or C.  A is not modified if FACT = 'F' or
              'N', or if FACT = 'E' and EQUED = 'N' on exit.
              On exit, if EQUED .ne. 'N', A is scaled as follows:
              EQUED = 'R':  A := diag(R) * A
              EQUED = 'C':  A := A * diag(C)
              EQUED = 'B':  A := diag(R) * A * diag(C).

    LDA

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

    AF

              AF is REAL array, dimension (LDAF,N)
              If FACT = 'F', then AF is an input argument and on entry
              contains the factors L and U from the factorization
              A = P*L*U as computed by SGETRF.  If EQUED .ne. 'N', then
              AF is the factored form of the equilibrated matrix A.
              If FACT = 'N', then AF is an output argument and on exit
              returns the factors L and U from the factorization A = P*L*U
              of the original matrix A.
              If FACT = 'E', then AF is an output argument and on exit
              returns the factors L and U from the factorization A = P*L*U
              of the equilibrated matrix A (see the description of A for
              the form of the equilibrated matrix).

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              If FACT = 'F', then IPIV is an input argument and on entry
              contains 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).
              If FACT = 'N', then IPIV is an output argument and on exit
              contains the pivot indices from the factorization A = P*L*U
              of the original matrix A.
              If FACT = 'E', then IPIV is an output argument and on exit
              contains the pivot indices from the factorization A = P*L*U
              of the equilibrated matrix A.

    EQUED

              EQUED is CHARACTER*1
              Specifies the form of equilibration that was done.
              = 'N':  No equilibration (always true if FACT = 'N').
              = '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).
              EQUED is an input argument if FACT = 'F'; otherwise, it is an
              output argument.

    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.  R is an input argument if FACT = 'F';
              otherwise, R is an output argument.  If FACT = 'F' and
              EQUED = 'R' or 'B', each element of R must be positive.

    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.  C is an input argument if FACT = 'F';
              otherwise, C is an output argument.  If FACT = 'F' and
              EQUED = 'C' or 'B', each element of C must be positive.

    B

              B is REAL array, dimension (LDB,NRHS)
              On entry, the N-by-NRHS right hand side matrix B.
              On exit,
              if EQUED = 'N', B is not modified;
              if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
              diag(R)*B;
              if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
              overwritten by diag(C)*B.

    LDB

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

    X

              X is REAL array, dimension (LDX,NRHS)
              If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
              to the original system of equations.  Note that A and B are
              modified on exit if EQUED .ne. 'N', and the solution to the
              equilibrated system is inv(diag(C))*X if TRANS = 'N' and
              EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
              and EQUED = 'R' or 'B'.

    LDX

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

    RCOND

              RCOND is REAL
              The estimate of the reciprocal condition number of the matrix
              A after equilibration (if done).  If RCOND is less than the
              machine precision (in particular, if RCOND = 0), the matrix
              is singular to working precision.  This condition is
              indicated by a return code of INFO > 0.

    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 (4*N)
              On exit, WORK(1) contains the reciprocal pivot growth
              factor norm(A)/norm(U). The "max absolute element" norm is
              used. If WORK(1) is much less than 1, then the stability
              of the LU factorization of the (equilibrated) matrix A
              could be poor. This also means that the solution X, condition
              estimator RCOND, and forward error bound FERR could be
              unreliable. If factorization fails with 0<INFO<=N, then
              WORK(1) contains the reciprocal pivot growth factor for the
              leading INFO columns of A.

    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
              > 0:  if INFO = i, and i is
                    <= N:  U(i,i) 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+1: U is nonsingular, but RCOND is less than machine
                           precision, meaning that the matrix is singular
                           to working precision.  Nevertheless, the
                           solution and error bounds are computed because
                           there are a number of situations where the
                           computed solution can be more accurate than the
                           value of RCOND would suggest.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

Definition at line 351 of file sgesvx.f.

  subroutine sgesvxx (character FACT, character TRANS, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, character EQUED, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , * ) X, integer LDX, real RCOND, real RPVGRW, 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)

SGESVXX computes the solution to system of linear equations A * X = B for GE matrices

Purpose:

        SGESVXX uses the LU factorization to compute the solution to a
        real system of linear equations  A * X = B,  where A is an
        N-by-N matrix and X and B are N-by-NRHS matrices.
        If requested, both normwise and maximum componentwise error bounds
        are returned. SGESVXX will return a solution with a tiny
        guaranteed error (O(eps) where eps is the working machine
        precision) unless the matrix is very ill-conditioned, in which
        case a warning is returned. Relevant condition numbers also are
        calculated and returned.
        SGESVXX accepts user-provided factorizations and equilibration
        factors; see the definitions of the FACT and EQUED options.
        Solving with refinement and using a factorization from a previous
        SGESVXX call will also produce a solution with either O(eps)
        errors or warnings, but we cannot make that claim for general
        user-provided factorizations and equilibration factors if they
        differ from what SGESVXX would itself produce.

Description:

        The following steps are performed:
        1. If FACT = 'E', real scaling factors are computed to equilibrate
        the system:
          TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
          TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
          TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
        Whether or not the system will be equilibrated depends on the
        scaling of the matrix A, but if equilibration is used, A is
        overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
        or diag(C)*B (if TRANS = 'T' or 'C').
        2. If FACT = 'N' or 'E', the LU decomposition is used to factor
        the matrix A (after equilibration if FACT = 'E') as
          A = P * L * U,
        where P is a permutation matrix, L is a unit lower triangular
        matrix, and U is upper triangular.
        3. If some U(i,i)=0, so that U is exactly singular, then the
        routine returns with INFO = i. Otherwise, the factored form of A
        is used to estimate the condition number of the matrix A (see
        argument RCOND). If the reciprocal of the condition number is less
        than machine precision, the routine still goes on to solve for X
        and compute error bounds as described below.
        4. The system of equations is solved for X using the factored form
        of A.
        5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
        the routine will use iterative refinement to try to get a small
        error and error bounds.  Refinement calculates the residual to at
        least twice the working precision.
        6. If equilibration was used, the matrix X is premultiplied by
        diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
        that it solves the original system before equilibration.

         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:
    FACT

              FACT is CHARACTER*1
         Specifies whether or not the factored form of the matrix A is
         supplied on entry, and if not, whether the matrix A should be
         equilibrated before it is factored.
           = 'F':  On entry, AF and IPIV contain the factored form of A.
                   If EQUED is not 'N', the matrix A has been
                   equilibrated with scaling factors given by R and C.
                   A, AF, and IPIV are not modified.
           = 'N':  The matrix A will be copied to AF and factored.
           = 'E':  The matrix A will be equilibrated if necessary, then
                   copied to AF and factored.

    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.

    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)
         On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
         not 'N', then A must have been equilibrated by the scaling
         factors in R and/or C.  A is not modified if FACT = 'F' or
         'N', or if FACT = 'E' and EQUED = 'N' on exit.
         On exit, if EQUED .ne. 'N', A is scaled as follows:
         EQUED = 'R':  A := diag(R) * A
         EQUED = 'C':  A := A * diag(C)
         EQUED = 'B':  A := diag(R) * A * diag(C).

    LDA

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

    AF

              AF is REAL array, dimension (LDAF,N)
         If FACT = 'F', then AF is an input argument and on entry
         contains the factors L and U from the factorization
         A = P*L*U as computed by SGETRF.  If EQUED .ne. 'N', then
         AF is the factored form of the equilibrated matrix A.
         If FACT = 'N', then AF is an output argument and on exit
         returns the factors L and U from the factorization A = P*L*U
         of the original matrix A.
         If FACT = 'E', then AF is an output argument and on exit
         returns the factors L and U from the factorization A = P*L*U
         of the equilibrated matrix A (see the description of A for
         the form of the equilibrated matrix).

    LDAF

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

    IPIV

              IPIV is INTEGER array, dimension (N)
         If FACT = 'F', then IPIV is an input argument and on entry
         contains 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).
         If FACT = 'N', then IPIV is an output argument and on exit
         contains the pivot indices from the factorization A = P*L*U
         of the original matrix A.
         If FACT = 'E', then IPIV is an output argument and on exit
         contains the pivot indices from the factorization A = P*L*U
         of the equilibrated matrix A.

    EQUED

              EQUED is CHARACTER*1
         Specifies the form of equilibration that was done.
           = 'N':  No equilibration (always true if FACT = 'N').
           = '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).
         EQUED is an input argument if FACT = 'F'; otherwise, it is an
         output argument.

    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.  R is an input argument if FACT = 'F';
         otherwise, R is an output argument.  If FACT = 'F' and
         EQUED = 'R' or 'B', each element of R must be positive.
         If R is output, each element of R is a power of the radix.
         If R is input, 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.  C is an input argument if FACT = 'F';
         otherwise, C is an output argument.  If FACT = 'F' and
         EQUED = 'C' or 'B', each element of C must be positive.
         If C is output, each element of C is a power of the radix.
         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)
         On entry, the N-by-NRHS right hand side matrix B.
         On exit,
         if EQUED = 'N', B is not modified;
         if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
            diag(R)*B;
         if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
            overwritten by diag(C)*B.

    LDB

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

    X

              X is REAL array, dimension (LDX,NRHS)
         If INFO = 0, the N-by-NRHS solution matrix X to the original
         system of equations.  Note that A and B are modified on exit
         if EQUED .ne. 'N', and the solution to the equilibrated system is
         inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
         inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.

    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.

    RPVGRW

              RPVGRW is REAL
         Reciprocal pivot growth.  On exit, this contains the reciprocal
         pivot growth factor norm(A)/norm(U). The "max absolute element"
         norm is used.  If this is much less than 1, then 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. If factorization fails with
         0<INFO<=N, then this contains the reciprocal pivot growth factor
         for the leading INFO columns of A.  In SGESVX, this quantity is
         returned in WORK(1).

    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:
    April 2012

Definition at line 545 of file sgesvxx.f.

Author

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