Provided by: liblapack-doc-man_3.5.0-2ubuntu1_all bug

NAME

       sla_gerfsx_extended.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV,
           COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL,
           RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
           SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for
           general matrices by performing extra-precise iterative refinement and provides error
           bounds and backward error estimates for the solution.

Function/Subroutine Documentation

   subroutine sla_gerfsx_extended (integerPREC_TYPE, integerTRANS_TYPE, integerN, integerNRHS,
       real, dimension( lda, * )A, integerLDA, real, dimension( ldaf, * )AF, integerLDAF,
       integer, dimension( * )IPIV, logicalCOLEQU, real, dimension( * )C, real, dimension( ldb, *
       )B, integerLDB, real, dimension( ldy, * )Y, integerLDY, real, dimension( * )BERR_OUT,
       integerN_NORMS, real, dimension( nrhs, * )ERRS_N, real, dimension( nrhs, * )ERRS_C, real,
       dimension( * )RES, real, dimension( * )AYB, real, dimension( * )DY, real, dimension( *
       )Y_TAIL, realRCOND, integerITHRESH, realRTHRESH, realDZ_UB, logicalIGNORE_CWISE,
       integerINFO)
       SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for
       general matrices by performing extra-precise iterative refinement and provides error
       bounds and backward error estimates for the solution.

       Purpose:

            SLA_GERFSX_EXTENDED improves the computed solution to a system of
            linear equations by performing extra-precise iterative refinement
            and provides error bounds and backward error estimates for the solution.
            This subroutine is called by SGERFSX to perform iterative refinement.
            In addition to normwise error bound, the code provides maximum
            componentwise error bound if possible. See comments for ERRS_N
            and ERRS_C for details of the error bounds. Note that this
            subroutine is only resonsible for setting the second fields of
            ERRS_N and ERRS_C.

       Parameters:
           PREC_TYPE

                     PREC_TYPE is INTEGER
                Specifies the intermediate precision to be used in refinement.
                The value is defined by ILAPREC(P) where P is a CHARACTER and
                P    = 'S':  Single
                     = 'D':  Double
                     = 'I':  Indigenous
                     = 'X', 'E':  Extra

           TRANS_TYPE

                     TRANS_TYPE is INTEGER
                Specifies the transposition operation on A.
                The value is defined by ILATRANS(T) where T is a CHARACTER and
                T    = 'N':  No transpose
                     = 'T':  Transpose
                     = 'C':  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
                matrix B.

           A

                     A is REAL array, dimension (LDA,N)
                On entry, the 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 the factorization A = P*L*U
                as computed by SGETRF; row i of the matrix was interchanged
                with row IPIV(i).

           COLEQU

                     COLEQU is LOGICAL
                If .TRUE. then column equilibration was done to A before calling
                this routine. This is needed to compute the solution and error
                bounds correctly.

           C

                     C is REAL array, dimension (N)
                The column scale factors for A. If COLEQU = .FALSE., C
                is not accessed. 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)
                The right-hand-side matrix B.

           LDB

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

           Y

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

           LDY

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

           BERR_OUT

                     BERR_OUT is REAL array, dimension (NRHS)
                On exit, BERR_OUT(j) contains the componentwise relative backward
                error for right-hand-side j from the formula
                    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
                where abs(Z) is the componentwise absolute value of the matrix
                or vector Z. This is computed by SLA_LIN_BERR.

           N_NORMS

                     N_NORMS is INTEGER
                Determines which error bounds to return (see ERRS_N
                and ERRS_C).
                If N_NORMS >= 1 return normwise error bounds.
                If N_NORMS >= 2 return componentwise error bounds.

           ERRS_N

                     ERRS_N 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 ERRS_N(i,:) corresponds to the ith
                right-hand side.

                The second index in ERRS_N(:,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.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           ERRS_C

                     ERRS_C 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
                ERRS_C is not accessed.  If N_ERR_BNDS .LT. 3, then at most
                the first (:,N_ERR_BNDS) entries are returned.

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

                The second index in ERRS_C(:,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.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           RES

                     RES is REAL array, dimension (N)
                Workspace to hold the intermediate residual.

           AYB

                     AYB is REAL array, dimension (N)
                Workspace. This can be the same workspace passed for Y_TAIL.

           DY

                     DY is REAL array, dimension (N)
                Workspace to hold the intermediate solution.

           Y_TAIL

                     Y_TAIL is REAL array, dimension (N)
                Workspace to hold the trailing bits of the intermediate solution.

           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.

           ITHRESH

                     ITHRESH is INTEGER
                The maximum number of residual computations allowed for
                refinement. The default is 10. For '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
                ERRS_N and ERRS_C may no longer be trustworthy.

           RTHRESH

                     RTHRESH is REAL
                Determines when to stop refinement if the error estimate stops
                decreasing. Refinement will stop when the next solution no longer
                satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
                the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
                default value is 0.5. For 'aggressive' set to 0.9 to permit
                convergence on extremely ill-conditioned matrices. See LAWN 165
                for more details.

           DZ_UB

                     DZ_UB is REAL
                Determines when to start considering componentwise convergence.
                Componentwise convergence is only considered after each component
                of the solution Y is stable, which we definte as the relative
                change in each component being less than DZ_UB. The default value
                is 0.25, requiring the first bit to be stable. See LAWN 165 for
                more details.

           IGNORE_CWISE

                     IGNORE_CWISE is LOGICAL
                If .TRUE. then ignore componentwise convergence. Default value
                is .FALSE..

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                  < 0:  if INFO = -i, the ith argument to SGETRS had an illegal
                        value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

       Definition at line 393 of file sla_gerfsx_extended.f.

Author

       Generated automatically by Doxygen for LAPACK from the source code.