Provided by: liblapack-doc_3.12.0-3build1.1_all bug

NAME

       gesvxx - gesvxx: factor and solve, extra precise

SYNOPSIS

   Functions
       subroutine cgesvxx (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, rwork, info)
            CGESVXX 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 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
       subroutine zgesvxx (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, rwork, info)
            ZGESVXX computes the solution to system of linear equations A * X = B for GE matrices

Detailed Description

Function Documentation

   subroutine cgesvxx (character fact, character trans, integer n, integer nrhs, complex,
       dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf,
       integer, dimension( * ) ipiv, character equed, real, dimension( * ) r, real, dimension( *
       ) c, complex, dimension( ldb, * ) b, integer ldb, complex, 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, complex, dimension( * ) work, real, dimension( * )
       rwork, integer info)
        CGESVXX computes the solution to system of linear equations A * X = B for GE matrices

       Purpose:

               CGESVXX uses the LU factorization to compute the solution to a
               complex 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. CGESVXX 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.

               CGESVXX 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
               CGESVXX 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 CGESVXX 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)

           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 COMPLEX 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 COMPLEX 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 CGETRF.  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 CGETRF; 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 COMPLEX 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 COMPLEX 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 CGESVX, 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 < 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 <= 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 < 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 COMPLEX array, dimension (2*N)

           RWORK

                     RWORK is REAL array, dimension (2*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 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 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 < 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 <= 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 < 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.

   subroutine zgesvxx (character fact, character trans, integer n, integer nrhs, complex*16,
       dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf,
       integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) r, double
       precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16,
       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, complex*16, dimension( * ) work, double
       precision, dimension( * ) rwork, integer info)
        ZGESVXX computes the solution to system of linear equations A * X = B for GE matrices

       Purpose:

               ZGESVXX uses the LU factorization to compute the solution to a
               complex*16 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. ZGESVXX 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.

               ZGESVXX 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
               ZGESVXX 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 ZGESVXX 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)

           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 COMPLEX*16 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 COMPLEX*16 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 ZGETRF.  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 ZGETRF; 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 COMPLEX*16 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 COMPLEX*16 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 ZGESVX, 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 COMPLEX*16 array, dimension (2*N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (2*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.

Author

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