GNU.WIKI: The GNU/Linux Knowledge Base

  [HOME] [PHP Manual] [HowTo] [ABS] [MAN1] [MAN2] [MAN3] [MAN4] [MAN5] [MAN6] [MAN7] [MAN8] [MAN9]

  [0-9] [Aa] [Bb] [Cc] [Dd] [Ee] [Ff] [Gg] [Hh] [Ii] [Jj] [Kk] [Ll] [Mm] [Nn] [Oo] [Pp] [Qq] [Rr] [Ss] [Tt] [Uu] [Vv] [Ww] [Xx] [Yy] [Zz]


NAME

       zla_gbrfsx_extended.f -

SYNOPSIS

   Functions/Subroutines
       subroutine zla_gbrfsx_extended (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS,
           AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT,
           N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND,
           ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
           ZLA_GBRFSX_EXTENDED improves the computed solution to a system of
           linear equations for general banded matrices by performing extra-
           precise iterative refinement and provides error bounds and backward
           error estimates for the solution.

Function/Subroutine Documentation

   subroutine zla_gbrfsx_extended (integerPREC_TYPE, integerTRANS_TYPE,
       integerN, integerKL, integerKU, integerNRHS, complex*16, dimension(
       ldab, * )AB, integerLDAB, complex*16, dimension( ldafb, * )AFB,
       integerLDAFB, integer, dimension( * )IPIV, logicalCOLEQU, double
       precision, dimension( * )C, complex*16, dimension( ldb, * )B,
       integerLDB, complex*16, dimension( ldy, * )Y, integerLDY, double
       precision, dimension( * )BERR_OUT, integerN_NORMS, double precision,
       dimension( nrhs, * )ERR_BNDS_NORM, double precision, dimension( nrhs, *
       )ERR_BNDS_COMP, complex*16, dimension( * )RES, double precision,
       dimension(*)AYB, complex*16, dimension( * )DY, complex*16, dimension( *
       )Y_TAIL, double precisionRCOND, integerITHRESH, double
       precisionRTHRESH, double precisionDZ_UB, logicalIGNORE_CWISE,
       integerINFO)
       ZLA_GBRFSX_EXTENDED improves the computed solution to a system of
       linear equations for general banded matrices by performing extra-
       precise iterative refinement and provides error bounds and backward
       error estimates for the solution.

       Purpose:

            ZLA_GBRFSX_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 ZGBRFSX to perform iterative refinement.
            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. Note that this
            subroutine is only resonsible for setting the second fields of
            ERR_BNDS_NORM and ERR_BNDS_COMP.

       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.

           KL

                     KL is INTEGER
                The number of subdiagonals within the band of A.  KL >= 0.

           KU

                     KU is INTEGER
                The number of superdiagonals within the band of A.  KU >= 0

           NRHS

                     NRHS is INTEGER
                The number of right-hand-sides, i.e., the number of columns of the
                matrix B.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB,N)
                On entry, the N-by-N matrix A.

           LDAB

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

           AFB

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

           LDAFB

                     LDAFB 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 ZGBTRF; 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 DOUBLE PRECISION 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 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).

           Y

                     Y is COMPLEX*16 array, dimension (LDY,NRHS)
                On entry, the solution matrix X, as computed by ZGBTRS.
                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 DOUBLE PRECISION 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 ZLA_LIN_BERR.

           N_NORMS

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

           ERR_BNDS_NORM

                     ERR_BNDS_NORM is DOUBLE PRECISION array, dimension
                               (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

                The array is indexed by the type of error information as described
                below. There currently are up to three pieces of information
                returned.

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

                The second index in ERR_BNDS_NORM(:,err) contains the following
                three fields:
                err = 1 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * 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.

           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 .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.

                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 COMPLEX*16 array, dimension (N)
                Workspace to hold the intermediate residual.

           AYB

                     AYB is DOUBLE PRECISION array, dimension (N)
                Workspace.

           DY

                     DY is COMPLEX*16 array, dimension (N)
                Workspace to hold the intermediate solution.

           Y_TAIL

                     Y_TAIL is COMPLEX*16 array, dimension (N)
                Workspace to hold the trailing bits of the intermediate solution.

           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.

           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
                ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

           RTHRESH

                     RTHRESH is DOUBLE PRECISION
                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 DOUBLE PRECISION
                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 ZGBTRS had an illegal
                        value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

       Definition at line 407 of file zla_gbrfsx_extended.f.

Author

       Generated automatically by Doxygen for LAPACK from the source code.



  All copyrights belong to their respective owners. Other content (c) 2014-2018, GNU.WIKI. Please report site errors to webmaster@gnu.wiki.
Page load time: 0.114 seconds. Last modified: November 04 2018 12:49:43.