Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  SGERFSX  improve  the  computed solution to a system of linear  equations and
       provides error bounds and backward error estimates  for the solution

SYNOPSIS

       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 )

           IMPLICIT        NONE

           CHARACTER       TRANS, EQUED

           INTEGER         INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, N_ERR_BNDS

           REAL            RCOND

           INTEGER         IPIV( * ), IWORK( * )

           REAL            A( LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX , * ), WORK( * )

           REAL            R( * ), C( * ), PARAMS( * ), BERR( *  ),  ERR_BNDS_NORM(  NRHS,  *  ),
                           ERR_BNDS_COMP( NRHS, * )

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.

ARGUMENTS

        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.

        TRANS   (input) 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   (input) 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       (input) INTEGER
                The order of the matrix A.  N >= 0.

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

        A       (input) REAL array, dimension (LDA,N)
                The original N-by-N matrix A.

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

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

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

        IPIV    (input) 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       (input) 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       (input) 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       (input) REAL array, dimension (LDB,NRHS)
                The right hand side matrix B.

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

        X       (input/output) REAL array, dimension (LDX,NRHS)
                On entry, the solution matrix X, as computed by SGETRS.
                On exit, the improved solution matrix X.

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

        RCOND   (output) 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    (output) 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 (input) 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  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
                       For each right-hand side, this array contains information about
                       various error bounds and condition numbers corresponding to the
                       normwise relative error, which is defined as follows:
                       Normwise relative error in the ith solution vector:
                       max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                       max_j abs(X(j,i))
                       The array is indexed by the type of error information as described
                       below. There currently are up to three pieces of information
                       returned.
                       The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                       right-hand side.
                       The second index in ERR_BNDS_NORM(:,err) contains the following
                       three fields:
                       err = 1 "Trust/don't trust" boolean. Trust the answer if the
                       reciprocal condition number is less than the threshold
                       sqrt(n) * slamch('Epsilon').
                       err = 2 "Guaranteed" error bound: The estimated forward error,
                       almost certainly within a factor of 10 of the true error
                       so long as the next entry is greater than the threshold
                       sqrt(n) * slamch('Epsilon'). This error bound should only
                       be trusted if the previous boolean is true.
                       err = 3  Reciprocal condition number: Estimated normwise
                       reciprocal condition number.  Compared with the threshold
                       sqrt(n) * slamch('Epsilon') to determine if the error
                       estimate is "guaranteed". These reciprocal condition
                       numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                       appropriately scaled matrix Z.
                       Let Z = S*A, where S scales each row by a power of the
                       radix so all absolute row sums of Z are approximately 1.
                       See Lapack Working Note 165 for further details and extra
                       cautions.

        ERR_BNDS_COMP  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
                       For each right-hand side, this array contains information about
                       various error bounds and condition numbers corresponding to the
                       componentwise relative error, which is defined as follows:
                       Componentwise relative error in the ith solution vector:
                       abs(XTRUE(j,i) - X(j,i))
                       max_j ----------------------
                       abs(X(j,i))
                       The array is indexed by the right-hand side i (on which the
                       componentwise relative error depends), and the type of error
                       information as described below. There currently are up to three
                       pieces of information returned for each right-hand side. If
                       componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                       ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
                       the first (:,N_ERR_BNDS) entries are returned.
                       The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                       right-hand side.
                       The second index in ERR_BNDS_COMP(:,err) contains the following
                       three fields:
                       err = 1 "Trust/don't trust" boolean. Trust the answer if the
                       reciprocal condition number is less than the threshold
                       sqrt(n) * slamch('Epsilon').
                       err = 2 "Guaranteed" error bound: The estimated forward error,
                       almost certainly within a factor of 10 of the true error
                       so long as the next entry is greater than the threshold
                       sqrt(n) * slamch('Epsilon'). This error bound should only
                       be trusted if the previous boolean is true.
                       err = 3  Reciprocal condition number: Estimated componentwise
                       reciprocal condition number.  Compared with the threshold
                       sqrt(n) * slamch('Epsilon') to determine if the error
                       estimate is "guaranteed". These reciprocal condition
                       numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                       appropriately scaled matrix Z.
                       Let Z = S*(A*diag(x)), where x is the solution for the
                       current right-hand side and S scales each row of
                       A*diag(x) by a power of the radix so all absolute row
                       sums of Z are approximately 1.
                       See Lapack Working Note 165 for further details and extra
                       cautions.
                       NPARAMS (input) INTEGER
                       Specifies the number of parameters set in PARAMS.  If .LE. 0, the
                       PARAMS array is never referenced and default values are used.

        PARAMS  (input / output) REAL array, dimension NPARAMS
                Specifies algorithm parameters.  If an entry is .LT. 0.0, then
                that entry will be filled with default value used for that
                parameter.  Only positions up to NPARAMS are accessed; defaults
                are used for higher-numbered parameters.
                PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                refinement or not.
                Default: 1.0
                = 0.0 : No refinement is performed, and no error bounds are
                computed.
                = 1.0 : Use the double-precision refinement algorithm,
                possibly with doubled-single computations if the
                compilation environment does not support DOUBLE
                PRECISION.
                (other values are reserved for future use)
                PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                computations allowed for refinement.
                Default: 10
                Aggressive: Set to 100 to permit convergence using approximate
                factorizations or factorizations other than LU. If
                the factorization uses a technique other than
                Gaussian elimination, the guarantees in
                err_bnds_norm and err_bnds_comp may no longer be
                trustworthy.
                PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                will attempt to find a solution with small componentwise
                relative error in the double-precision algorithm.  Positive
                is true, 0.0 is false.
                Default: 1.0 (attempt componentwise convergence)

        WORK    (workspace) REAL array, dimension (4*N)

        IWORK   (workspace) INTEGER array, dimension (N)

        INFO    (output) 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.

    LAPACK routine (version 3.2.2)          April 2011                           SGERFSX(3lapack)