gesvxx(3)

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
    !>     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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  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 . 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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  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 . 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
    !>     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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  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 . 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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  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 . 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
    !>     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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  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 . 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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  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 . 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
    !>     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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  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 . 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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  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 . 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

Generated automatically by Doxygen for LAPACK from the source code.