Provided by: liblapack-doc_3.11.0-2build1_all bug

NAME

       doubleGEsolve - double

SYNOPSIS

   Functions
       subroutine dgels (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO)
            DGELS solves overdetermined or underdetermined systems for GE matrices
       subroutine dgelsd (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO)
            DGELSD computes the minimum-norm solution to a linear least squares problem for GE
           matrices
       subroutine dgelss (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO)
            DGELSS solves overdetermined or underdetermined systems for GE matrices
       subroutine dgelst (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO)
            DGELST solves overdetermined or underdetermined systems for GE matrices using QR or
           LQ factorization with compact WY representation of Q.
       subroutine dgelsy (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO)
            DGELSY solves overdetermined or underdetermined systems for GE matrices
       subroutine dgesv (N, NRHS, A, LDA, IPIV, B, LDB, INFO)
            DGESV computes the solution to system of linear equations A * X = B for GE matrices
       subroutine dgesvx (FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X,
           LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
            DGESVX computes the solution to system of linear equations A * X = B for GE matrices
       subroutine dgesvxx (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)
            DGESVXX computes the solution to system of linear equations A * X = B for GE matrices
       subroutine dgetsls (TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO)
           DGETSLS
       subroutine dsgesv (N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, SWORK, ITER, INFO)
            DSGESV computes the solution to system of linear equations A * X = B for GE matrices
           (mixed precision with iterative refinement)

Detailed Description

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

Function Documentation

   subroutine dgels (character TRANS, integer M, integer N, integer NRHS, double precision,
       dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB,
       double precision, dimension( * ) WORK, integer LWORK, integer INFO)
        DGELS solves overdetermined or underdetermined systems for GE matrices

       Purpose:

            DGELS 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 DOUBLE PRECISION 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 DGEQRF;
                       if M <  N, A is overwritten by details of its LQ
                                  factorization as returned by DGELQF.

           LDA

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

           B

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

           LWORK

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

   subroutine dgelsd (integer M, integer N, integer NRHS, double precision, dimension( lda, * )
       A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision,
       dimension( * ) S, double precision RCOND, integer RANK, double precision, dimension( * )
       WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)
        DGELSD computes the minimum-norm solution to a linear least squares problem for GE
       matrices

       Purpose:

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

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK 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 WORK array, returns
                     this value as the first entry of the WORK array, 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.

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

   subroutine dgelss (integer M, integer N, integer NRHS, double precision, dimension( lda, * )
       A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision,
       dimension( * ) S, double precision RCOND, integer RANK, double precision, dimension( * )
       WORK, integer LWORK, integer INFO)
        DGELSS solves overdetermined or underdetermined systems for GE matrices

       Purpose:

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

           LWORK

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

   subroutine dgelst (character TRANS, integer M, integer N, integer NRHS, double precision,
       dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB,
       double precision, dimension( * ) WORK, integer LWORK, integer INFO)
        DGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ
       factorization with compact WY representation of Q.

       Purpose:

            DGELST 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 with compact WY representation of Q.
            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 DOUBLE PRECISION 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 DGEQRT;
                       if M <  N, A is overwritten by details of its LQ
                                  factorization as returned by DGELQT.

           LDA

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

           B

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

           LWORK

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

       Contributors:

             November 2022,  Igor Kozachenko,
                             Computer Science Division,
                             University of California, Berkeley

   subroutine dgelsy (integer M, integer N, integer NRHS, double precision, dimension( lda, * )
       A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer, dimension(
       * ) JPVT, double precision RCOND, integer RANK, double precision, dimension( * ) WORK,
       integer LWORK, integer INFO)
        DGELSY solves overdetermined or underdetermined systems for GE matrices

       Purpose:

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

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     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 DGEQP3, DTZRZF, STZRQF, DORMQR,
                     and DORMRZ.

                     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.

       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

   subroutine dgesv (integer N, integer NRHS, double precision, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB,
       integer INFO)
        DGESV computes the solution to system of linear equations A * X = B for GE matrices

       Purpose:

            DGESV 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

   subroutine dgesvx (character FACT, character TRANS, integer N, integer NRHS, double precision,
       dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer
       LDAF, integer, dimension( * ) IPIV, character EQUED, double precision, dimension( * ) R,
       double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB,
       double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double
       precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision,
       dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
        DGESVX computes the solution to system of linear equations A * X = B for GE matrices

       Purpose:

            DGESVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DGETRF.  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 DGETRF; 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 DOUBLE PRECISION array, dimension (N)
                     The row scale factors for A.  If EQUED = 'R' or 'B', A is
                     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                     is not accessed.  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 DOUBLE PRECISION array, dimension (N)
                     The column scale factors for A.  If EQUED = 'C' or 'B', A is
                     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                     is not accessed.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
                     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 DOUBLE PRECISION array, dimension (NRHS)
                     The estimated forward error bound for each solution vector
                     X(j) (the j-th column of the solution matrix X).
                     If XTRUE is the true solution corresponding to X(j), FERR(j)
                     is an estimated upper bound for the magnitude of the largest
                     element in (X(j) - XTRUE) divided by the magnitude of the
                     largest element in X(j).  The estimate is as reliable as
                     the estimate for RCOND, and is almost always a slight
                     overestimate of the true error.

           BERR

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (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.

   subroutine dgesvxx (character FACT, character TRANS, integer N, integer NRHS, double
       precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF,
       integer LDAF, integer, dimension( * ) IPIV, character EQUED, double precision, dimension(
       * ) R, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B,
       integer LDB, double precision, dimension( ldx , * ) X, integer LDX, double precision
       RCOND, double precision RPVGRW, double precision, dimension( * ) BERR, integer N_ERR_BNDS,
       double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, *
       ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, double
       precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
        DGESVXX computes the solution to system of linear equations A * X = B for GE matrices

       Purpose:

               DGESVXX uses the LU factorization to compute the solution to a
               double precision 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. DGESVXX 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.

               DGESVXX 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
               DGESVXX 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 DGESVXX would itself produce.

       Description:

               The following steps are performed:

               1. If FACT = 'E', double precision 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DGETRF.  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 DGETRF; 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 DOUBLE PRECISION array, dimension (N)
                The row scale factors for A.  If EQUED = 'R' or 'B', A is
                multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                is not accessed.  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 DOUBLE PRECISION array, dimension (N)
                The column scale factors for A.  If EQUED = 'C' or 'B', A is
                multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                is not accessed.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
                Reciprocal scaled condition number.  This is an estimate of the
                reciprocal Skeel condition number of the matrix A after
                equilibration (if done).  If this is less than the machine
                precision (in particular, if it is zero), the matrix is singular
                to working precision.  Note that the error may still be small even
                if this number is very small and the matrix appears ill-
                conditioned.

           RPVGRW

                     RPVGRW is DOUBLE PRECISION
                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 DGESVX, this quantity is
                returned in WORK(1).

           BERR

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

           N_ERR_BNDS

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

           ERR_BNDS_NORM

                     ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

                The array is indexed by the type of error information as described
                below. There currently are up to three pieces of information
                returned.

                The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_NORM(:,err) contains the following
                three fields:
                err = 1 'Trust/don't trust' boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * dlamch('Epsilon').

                err = 2 'Guaranteed' error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * dlamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated normwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * dlamch('Epsilon') to determine if the error
                         estimate is 'guaranteed'. These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*A, where S scales each row by a power of the
                         radix so all absolute row sums of Z are approximately 1.

                See Lapack Working Note 165 for further details and extra
                cautions.

           ERR_BNDS_COMP

                     ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

                The array is indexed by the right-hand side i (on which the
                componentwise relative error depends), and the type of error
                information as described below. There currently are up to three
                pieces of information returned for each right-hand side. If
                componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
                the first (:,N_ERR_BNDS) entries are returned.

                The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_COMP(:,err) contains the following
                three fields:
                err = 1 'Trust/don't trust' boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * dlamch('Epsilon').

                err = 2 'Guaranteed' error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * dlamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated componentwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * dlamch('Epsilon') to determine if the error
                         estimate is 'guaranteed'. These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*(A*diag(x)), where x is the solution for the
                         current right-hand side and S scales each row of
                         A*diag(x) by a power of the radix so all absolute row
                         sums of Z are approximately 1.

                See Lapack Working Note 165 for further details and extra
                cautions.

           NPARAMS

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

           PARAMS

                     PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
                Specifies algorithm parameters.  If an entry is < 0.0, then
                that entry will be filled with default value used for that
                parameter.  Only positions up to NPARAMS are accessed; defaults
                are used for higher-numbered parameters.

                  PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                       refinement or not.
                    Default: 1.0D+0
                       = 0.0:  No refinement is performed, and no error bounds are
                               computed.
                       = 1.0:  Use the extra-precise refinement algorithm.
                         (other values are reserved for future use)

                  PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                       computations allowed for refinement.
                    Default: 10
                    Aggressive: Set to 100 to permit convergence using approximate
                                factorizations or factorizations other than LU. If
                                the factorization uses a technique other than
                                Gaussian elimination, the guarantees in
                                err_bnds_norm and err_bnds_comp may no longer be
                                trustworthy.

                  PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                       will attempt to find a solution with small componentwise
                       relative error in the double-precision algorithm.  Positive
                       is true, 0.0 is false.
                    Default: 1.0 (attempt componentwise convergence)

           WORK

                     WORK is DOUBLE PRECISION array, dimension (4*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dgetsls (character TRANS, integer M, integer N, integer NRHS, double precision,
       dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB,
       double precision, dimension( * ) WORK, integer LWORK, integer INFO)
       DGETSLS

       Purpose:

            DGETSLS solves overdetermined or underdetermined real linear systems
            involving an M-by-N matrix A, using a tall skinny QR or short wide 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 undetermined 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 DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit,
                     A is overwritten by details of its QR or LQ
                     factorization as returned by DGEQR or DGELQ.

           LDA

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

           B

                     B is DOUBLE PRECISION 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.
                     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.

           LDB

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

           WORK

                     (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
                     or optimal, if query was assumed) LWORK.
                     See LWORK for details.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If LWORK = -1 or -2, then a workspace query is assumed.
                     If LWORK = -1, the routine calculates optimal size of WORK for the
                     optimal performance and returns this value in WORK(1).
                     If LWORK = -2, the routine calculates minimal size of WORK and
                     returns this value in WORK(1).

           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.

   subroutine dsgesv (integer N, integer NRHS, double precision, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB,
       double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( n, * )
       WORK, real, dimension( * ) SWORK, integer ITER, integer INFO)
        DSGESV computes the solution to system of linear equations A * X = B for GE matrices
       (mixed precision with iterative refinement)

       Purpose:

            DSGESV 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.

            DSGESV first attempts to factorize the matrix in SINGLE PRECISION
            and use this factorization within an iterative refinement procedure
            to produce a solution with DOUBLE PRECISION normwise backward error
            quality (see below). If the approach fails the method switches to a
            DOUBLE PRECISION factorization and solve.

            The iterative refinement is not going to be a winning strategy if
            the ratio SINGLE PRECISION performance over DOUBLE PRECISION
            performance is too small. A reasonable strategy should take the
            number of right-hand sides and the size of the matrix into account.
            This might be done with a call to ILAENV in the future. Up to now, we
            always try iterative refinement.

            The iterative refinement process is stopped if
                ITER > ITERMAX
            or for all the RHS we have:
                RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
            where
                o ITER is the number of the current iteration in the iterative
                  refinement process
                o RNRM is the infinity-norm of the residual
                o XNRM is the infinity-norm of the solution
                o ANRM is the infinity-operator-norm of the matrix A
                o EPS is the machine epsilon returned by DLAMCH('Epsilon')
            The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
            respectively.

       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 DOUBLE PRECISION array,
                     dimension (LDA,N)
                     On entry, the N-by-N coefficient matrix A.
                     On exit, if iterative refinement has been successfully used
                     (INFO = 0 and ITER >= 0, see description below), then A is
                     unchanged, if double precision factorization has been used
                     (INFO = 0 and ITER < 0, see description below), then the
                     array A contains 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).
                     Corresponds either to the single precision factorization
                     (if INFO = 0 and ITER >= 0) or the double precision
                     factorization (if INFO = 0 and ITER < 0).

           B

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

           LDB

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

           X

                     X is DOUBLE PRECISION array, dimension (LDX,NRHS)
                     If INFO = 0, the N-by-NRHS solution matrix X.

           LDX

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (N,NRHS)
                     This array is used to hold the residual vectors.

           SWORK

                     SWORK is REAL array, dimension (N*(N+NRHS))
                     This array is used to use the single precision matrix and the
                     right-hand sides or solutions in single precision.

           ITER

                     ITER is INTEGER
                     < 0: iterative refinement has failed, double precision
                          factorization has been performed
                          -1 : the routine fell back to full precision for
                               implementation- or machine-specific reasons
                          -2 : narrowing the precision induced an overflow,
                               the routine fell back to full precision
                          -3 : failure of SGETRF
                          -31: stop the iterative refinement after the 30th
                               iterations
                     > 0: iterative refinement has been successfully used.
                          Returns the number of iterations

           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) computed in DOUBLE PRECISION 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.

Author

       Generated automatically by Doxygen for LAPACK from the source code.