gerfsx(3)

NAME
gerfsx - gerfsx: iterative refinement, expert

SYNOPSIS

  Functions

subroutine cgerfsx (trans, equed, n, nrhs, a, lda, af, ldaf, ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CGERFSX
subroutine dgerfsx (trans, equed, n, nrhs, a, lda, af, ldaf, ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
DGERFSX
subroutine sgerfsx (trans, equed, n, nrhs, a, lda, af, ldaf, ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
SGERFSX
subroutine zgerfsx (trans, equed, n, nrhs, a, lda, af, ldaf, ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
ZGERFSX

Detailed Description

Function Documentation

  subroutine cgerfsx (character trans, character equed, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real, dimension( * ) r, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx , * ) x, integer ldx, real rcond, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGERFSX

Purpose:

    !>
    !>    CGERFSX improves the computed solution to a system of linear
    !>    equations and provides error bounds and backward error estimates
    !>    for the solution.  In addition to normwise error bound, the code
    !>    provides maximum componentwise error bound if possible.  See
    !>    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
    !>    error bounds.
    !>
    !>    The original system of linear equations may have been equilibrated
    !>    before calling this routine, as described by arguments EQUED, R
    !>    and C below. In this case, the solution and error bounds returned
    !>    are for the original unequilibrated system.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>     Specifies the form of the system of equations:
    !>       = 'N':  A * X = B     (No transpose)
    !>       = 'T':  A**T * X = B  (Transpose)
    !>       = 'C':  A**H * X = B  (Conjugate transpose)
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>     Specifies the form of equilibration that was done to A
    !>     before calling this routine. This is needed to compute
    !>     the solution and error bounds correctly.
    !>       = 'N':  No equilibration
    !>       = 'R':  Row equilibration, i.e., A has been premultiplied by
    !>               diag(R).
    !>       = 'C':  Column equilibration, i.e., A has been postmultiplied
    !>               by diag(C).
    !>       = 'B':  Both row and column equilibration, i.e., A has been
    !>               replaced by diag(R) * A * diag(C).
    !>               The right hand side B has been changed accordingly.
    !>

    N

    !>          N is INTEGER
    !>     The order of the matrix A.  N >= 0.
    !>

    NRHS

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

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>     The original N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is COMPLEX array, dimension (LDAF,N)
    !>     The factors L and U from the factorization A = P*L*U
    !>     as computed by CGETRF.
    !>

    LDAF

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

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>     The pivot indices from CGETRF; for 1<=i<=N, row i of the
    !>     matrix was interchanged with row IPIV(i).
    !>

    R

    !>          R is REAL array, dimension (N)
    !>     The row scale factors for A.  If EQUED = 'R' or 'B', A is
    !>     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
    !>     is not accessed.
    !>     If R is accessed, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    C

    !>          C is REAL array, dimension (N)
    !>     The column scale factors for A.  If EQUED = 'C' or 'B', A is
    !>     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
    !>     is not accessed.
    !>     If C is accessed, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,NRHS)
    !>     The right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is COMPLEX array, dimension (LDX,NRHS)
    !>     On entry, the solution matrix X, as computed by CGETRS.
    !>     On exit, the improved solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is REAL
    !>     Reciprocal scaled condition number.  This is an estimate of the
    !>     reciprocal Skeel condition number of the matrix A after
    !>     equilibration (if done).  If this is less than the machine
    !>     precision (in particular, if it is zero), the matrix is singular
    !>     to working precision.  Note that the error may still be small even
    !>     if this number is very small and the matrix appears ill-
    !>     conditioned.
    !>

    BERR

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

    N_ERR_BNDS

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

    ERR_BNDS_NORM

    !>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     normwise relative error, which is defined as follows:
    !>
    !>     Normwise relative error in the ith solution vector:
    !>             max_j (abs(XTRUE(j,i) - X(j,i)))
    !>            ------------------------------
    !>                  max_j abs(X(j,i))
    !>
    !>     The array is indexed by the type of error information as described
    !>     below. There currently are up to three pieces of information
    !>     returned.
    !>
    !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_NORM(:,err) contains the following
    !>     three fields:
    !>     err = 1  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 dgerfsx (character trans, character equed, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, double precision, dimension( * ) r, double precision, dimension( * ) c, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx , * ) x, integer ldx, double precision rcond, double precision, dimension( * ) 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)

DGERFSX

Purpose:

    !>
    !>    DGERFSX improves the computed solution to a system of linear
    !>    equations and provides error bounds and backward error estimates
    !>    for the solution.  In addition to normwise error bound, the code
    !>    provides maximum componentwise error bound if possible.  See
    !>    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
    !>    error bounds.
    !>
    !>    The original system of linear equations may have been equilibrated
    !>    before calling this routine, as described by arguments EQUED, R
    !>    and C below. In this case, the solution and error bounds returned
    !>    are for the original unequilibrated system.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>     Specifies the form of the system of equations:
    !>       = 'N':  A * X = B     (No transpose)
    !>       = 'T':  A**T * X = B  (Transpose)
    !>       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>     Specifies the form of equilibration that was done to A
    !>     before calling this routine. This is needed to compute
    !>     the solution and error bounds correctly.
    !>       = 'N':  No equilibration
    !>       = 'R':  Row equilibration, i.e., A has been premultiplied by
    !>               diag(R).
    !>       = 'C':  Column equilibration, i.e., A has been postmultiplied
    !>               by diag(C).
    !>       = 'B':  Both row and column equilibration, i.e., A has been
    !>               replaced by diag(R) * A * diag(C).
    !>               The right hand side B has been changed accordingly.
    !>

    N

    !>          N is INTEGER
    !>     The order of the matrix A.  N >= 0.
    !>

    NRHS

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

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>     The original N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
    !>     The factors L and U from the factorization A = P*L*U
    !>     as computed by DGETRF.
    !>

    LDAF

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

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>     The pivot indices from DGETRF; for 1<=i<=N, row i of the
    !>     matrix was interchanged with row IPIV(i).
    !>

    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.
    !>     If R is accessed, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    C

    !>          C is 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.
    !>     If C is accessed, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
    !>     The right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
    !>     On entry, the solution matrix X, as computed by DGETRS.
    !>     On exit, the improved solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is 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.
    !>

    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 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 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 sgerfsx (character trans, character equed, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, real, dimension( * ) r, real, dimension( * ) c, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx , * ) x, integer ldx, real rcond, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGERFSX

Purpose:

    !>
    !>    SGERFSX improves the computed solution to a system of linear
    !>    equations and provides error bounds and backward error estimates
    !>    for the solution.  In addition to normwise error bound, the code
    !>    provides maximum componentwise error bound if possible.  See
    !>    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
    !>    error bounds.
    !>
    !>    The original system of linear equations may have been equilibrated
    !>    before calling this routine, as described by arguments EQUED, R
    !>    and C below. In this case, the solution and error bounds returned
    !>    are for the original unequilibrated system.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>     Specifies the form of the system of equations:
    !>       = 'N':  A * X = B     (No transpose)
    !>       = 'T':  A**T * X = B  (Transpose)
    !>       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>     Specifies the form of equilibration that was done to A
    !>     before calling this routine. This is needed to compute
    !>     the solution and error bounds correctly.
    !>       = 'N':  No equilibration
    !>       = 'R':  Row equilibration, i.e., A has been premultiplied by
    !>               diag(R).
    !>       = 'C':  Column equilibration, i.e., A has been postmultiplied
    !>               by diag(C).
    !>       = 'B':  Both row and column equilibration, i.e., A has been
    !>               replaced by diag(R) * A * diag(C).
    !>               The right hand side B has been changed accordingly.
    !>

    N

    !>          N is INTEGER
    !>     The order of the matrix A.  N >= 0.
    !>

    NRHS

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

    A

    !>          A is REAL array, dimension (LDA,N)
    !>     The original N-by-N matrix A.
    !>

    LDA

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

    AF

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

    LDAF

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

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>     The pivot indices from SGETRF; for 1<=i<=N, row i of the
    !>     matrix was interchanged with row IPIV(i).
    !>

    R

    !>          R is REAL array, dimension (N)
    !>     The row scale factors for A.  If EQUED = 'R' or 'B', A is
    !>     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
    !>     is not accessed.
    !>     If R is accessed, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    C

    !>          C is REAL array, dimension (N)
    !>     The column scale factors for A.  If EQUED = 'C' or 'B', A is
    !>     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
    !>     is not accessed.
    !>     If C is accessed, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

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

    LDB

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

    X

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

    LDX

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

    RCOND

    !>          RCOND is REAL
    !>     Reciprocal scaled condition number.  This is an estimate of the
    !>     reciprocal Skeel condition number of the matrix A after
    !>     equilibration (if done).  If this is less than the machine
    !>     precision (in particular, if it is zero), the matrix is singular
    !>     to working precision.  Note that the error may still be small even
    !>     if this number is very small and the matrix appears ill-
    !>     conditioned.
    !>

    BERR

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

    N_ERR_BNDS

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

    ERR_BNDS_NORM

    !>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     normwise relative error, which is defined as follows:
    !>
    !>     Normwise relative error in the ith solution vector:
    !>             max_j (abs(XTRUE(j,i) - X(j,i)))
    !>            ------------------------------
    !>                  max_j abs(X(j,i))
    !>
    !>     The array is indexed by the type of error information as described
    !>     below. There currently are up to three pieces of information
    !>     returned.
    !>
    !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_NORM(:,err) contains the following
    !>     three fields:
    !>     err = 1  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 zgerfsx (character trans, character equed, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, 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, 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)

ZGERFSX

Purpose:

    !>
    !>    ZGERFSX improves the computed solution to a system of linear
    !>    equations and provides error bounds and backward error estimates
    !>    for the solution.  In addition to normwise error bound, the code
    !>    provides maximum componentwise error bound if possible.  See
    !>    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
    !>    error bounds.
    !>
    !>    The original system of linear equations may have been equilibrated
    !>    before calling this routine, as described by arguments EQUED, R
    !>    and C below. In this case, the solution and error bounds returned
    !>    are for the original unequilibrated system.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>     Specifies the form of the system of equations:
    !>       = 'N':  A * X = B     (No transpose)
    !>       = 'T':  A**T * X = B  (Transpose)
    !>       = 'C':  A**H * X = B  (Conjugate transpose)
    !>

    EQUED

    !>          EQUED is CHARACTER*1
    !>     Specifies the form of equilibration that was done to A
    !>     before calling this routine. This is needed to compute
    !>     the solution and error bounds correctly.
    !>       = 'N':  No equilibration
    !>       = 'R':  Row equilibration, i.e., A has been premultiplied by
    !>               diag(R).
    !>       = 'C':  Column equilibration, i.e., A has been postmultiplied
    !>               by diag(C).
    !>       = 'B':  Both row and column equilibration, i.e., A has been
    !>               replaced by diag(R) * A * diag(C).
    !>               The right hand side B has been changed accordingly.
    !>

    N

    !>          N is INTEGER
    !>     The order of the matrix A.  N >= 0.
    !>

    NRHS

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

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>     The original N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is COMPLEX*16 array, dimension (LDAF,N)
    !>     The factors L and U from the factorization A = P*L*U
    !>     as computed by ZGETRF.
    !>

    LDAF

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

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>     The pivot indices from ZGETRF; for 1<=i<=N, row i of the
    !>     matrix was interchanged with row IPIV(i).
    !>

    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.
    !>     If R is accessed, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    C

    !>          C is 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.
    !>     If C is accessed, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB,NRHS)
    !>     The right hand side matrix B.
    !>

    LDB

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

    X

    !>          X is COMPLEX*16 array, dimension (LDX,NRHS)
    !>     On entry, the solution matrix X, as computed by ZGETRS.
    !>     On exit, the improved solution matrix X.
    !>

    LDX

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

    RCOND

    !>          RCOND is 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.
    !>

    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 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*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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