version 1.1, 2010/08/07 13:21:05
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version 1.14, 2018/05/29 07:18:04
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*> \brief \b DPORFSX |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DPORFSX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dporfsx.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dporfsx.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dporfsx.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, |
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* LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, |
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* ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, |
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* WORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER UPLO, EQUED |
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* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, |
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* $ N_ERR_BNDS |
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* DOUBLE PRECISION RCOND |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IWORK( * ) |
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* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), |
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* $ X( LDX, * ), WORK( * ) |
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* DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), |
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* $ ERR_BNDS_NORM( NRHS, * ), |
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* $ ERR_BNDS_COMP( NRHS, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DPORFSX improves the computed solution to a system of linear |
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*> equations when the coefficient matrix is symmetric positive |
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*> definite, and provides error bounds and backward error estimates |
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*> for the solution. In addition to normwise error bound, the code |
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*> provides maximum componentwise error bound if possible. See |
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*> comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the |
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*> error bounds. |
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*> |
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*> The original system of linear equations may have been equilibrated |
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*> before calling this routine, as described by arguments EQUED and S |
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*> below. In this case, the solution and error bounds returned are |
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*> for the original unequilibrated system. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \verbatim |
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*> Some optional parameters are bundled in the PARAMS array. These |
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*> settings determine how refinement is performed, but often the |
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*> defaults are acceptable. If the defaults are acceptable, users |
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*> can pass NPARAMS = 0 which prevents the source code from accessing |
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*> the PARAMS argument. |
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*> \endverbatim |
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*> |
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*> \param[in] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> = 'U': Upper triangle of A is stored; |
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*> = 'L': Lower triangle of A is stored. |
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*> \endverbatim |
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*> |
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*> \param[in] EQUED |
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*> \verbatim |
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*> EQUED is CHARACTER*1 |
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*> Specifies the form of equilibration that was done to A |
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*> before calling this routine. This is needed to compute |
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*> the solution and error bounds correctly. |
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*> = 'N': No equilibration |
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*> = 'Y': Both row and column equilibration, i.e., A has been |
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*> replaced by diag(S) * A * diag(S). |
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*> The right hand side B has been changed accordingly. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] NRHS |
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*> \verbatim |
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*> NRHS is INTEGER |
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*> The number of right hand sides, i.e., the number of columns |
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*> of the matrices B and X. NRHS >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension (LDA,N) |
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*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N |
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*> upper triangular part of A contains the upper triangular part |
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*> of the matrix A, and the strictly lower triangular part of A |
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*> is not referenced. If UPLO = 'L', the leading N-by-N lower |
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*> triangular part of A contains the lower triangular part of |
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*> the matrix A, and the strictly upper triangular part of A is |
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*> not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in] AF |
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*> \verbatim |
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*> AF is DOUBLE PRECISION array, dimension (LDAF,N) |
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*> The triangular factor U or L from the Cholesky factorization |
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*> A = U**T*U or A = L*L**T, as computed by DPOTRF. |
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*> \endverbatim |
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*> |
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*> \param[in] LDAF |
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*> \verbatim |
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*> LDAF is INTEGER |
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*> The leading dimension of the array AF. LDAF >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in,out] S |
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*> \verbatim |
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*> S is DOUBLE PRECISION array, dimension (N) |
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*> The row scale factors for A. If EQUED = 'Y', A is multiplied on |
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*> the left and right by diag(S). S is an input argument if FACT = |
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*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED |
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*> = 'Y', each element of S must be positive. If S is output, each |
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*> element of S is a power of the radix. If S is input, each element |
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*> of S should be a power of the radix to ensure a reliable solution |
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*> and error estimates. Scaling by powers of the radix does not cause |
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*> rounding errors unless the result underflows or overflows. |
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*> Rounding errors during scaling lead to refining with a matrix that |
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*> is not equivalent to the input matrix, producing error estimates |
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*> that may not be reliable. |
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*> \endverbatim |
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*> |
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*> \param[in] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) |
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*> The right hand side matrix B. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in,out] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) |
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*> On entry, the solution matrix X, as computed by DGETRS. |
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*> On exit, the improved solution matrix X. |
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*> \endverbatim |
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*> |
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*> \param[in] LDX |
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*> \verbatim |
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*> LDX is INTEGER |
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*> The leading dimension of the array X. LDX >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] RCOND |
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*> \verbatim |
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*> RCOND is DOUBLE PRECISION |
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*> Reciprocal scaled condition number. This is an estimate of the |
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*> reciprocal Skeel condition number of the matrix A after |
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*> equilibration (if done). If this is less than the machine |
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*> precision (in particular, if it is zero), the matrix is singular |
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*> to working precision. Note that the error may still be small even |
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*> if this number is very small and the matrix appears ill- |
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*> conditioned. |
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*> \endverbatim |
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*> |
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*> \param[out] BERR |
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*> \verbatim |
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*> BERR is DOUBLE PRECISION array, dimension (NRHS) |
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*> Componentwise relative backward error. This is the |
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*> componentwise relative backward error of each solution vector X(j) |
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*> (i.e., the smallest relative change in any element of A or B that |
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*> makes X(j) an exact solution). |
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*> \endverbatim |
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*> |
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*> \param[in] N_ERR_BNDS |
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*> \verbatim |
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*> N_ERR_BNDS is INTEGER |
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*> Number of error bounds to return for each right hand side |
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*> and each type (normwise or componentwise). See ERR_BNDS_NORM and |
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*> ERR_BNDS_COMP below. |
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*> \endverbatim |
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*> |
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*> \param[out] ERR_BNDS_NORM |
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*> \verbatim |
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*> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) |
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*> For each right-hand side, this array contains information about |
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*> various error bounds and condition numbers corresponding to the |
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*> normwise relative error, which is defined as follows: |
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*> |
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*> Normwise relative error in the ith solution vector: |
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*> max_j (abs(XTRUE(j,i) - X(j,i))) |
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*> ------------------------------ |
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*> max_j abs(X(j,i)) |
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*> |
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*> The array is indexed by the type of error information as described |
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*> below. There currently are up to three pieces of information |
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*> returned. |
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*> |
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*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith |
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*> right-hand side. |
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*> |
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*> The second index in ERR_BNDS_NORM(:,err) contains the following |
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*> three fields: |
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*> err = 1 "Trust/don't trust" boolean. Trust the answer if the |
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*> reciprocal condition number is less than the threshold |
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*> sqrt(n) * dlamch('Epsilon'). |
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*> |
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*> err = 2 "Guaranteed" error bound: The estimated forward error, |
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*> almost certainly within a factor of 10 of the true error |
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*> so long as the next entry is greater than the threshold |
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*> sqrt(n) * dlamch('Epsilon'). This error bound should only |
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*> be trusted if the previous boolean is true. |
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*> |
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*> err = 3 Reciprocal condition number: Estimated normwise |
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*> reciprocal condition number. Compared with the threshold |
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*> sqrt(n) * dlamch('Epsilon') to determine if the error |
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*> estimate is "guaranteed". These reciprocal condition |
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*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some |
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*> appropriately scaled matrix Z. |
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*> Let Z = S*A, where S scales each row by a power of the |
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*> radix so all absolute row sums of Z are approximately 1. |
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*> |
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*> See Lapack Working Note 165 for further details and extra |
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*> cautions. |
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*> \endverbatim |
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*> |
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*> \param[out] ERR_BNDS_COMP |
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*> \verbatim |
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*> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) |
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*> For each right-hand side, this array contains information about |
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*> various error bounds and condition numbers corresponding to the |
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*> componentwise relative error, which is defined as follows: |
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*> |
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*> Componentwise relative error in the ith solution vector: |
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*> abs(XTRUE(j,i) - X(j,i)) |
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*> max_j ---------------------- |
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*> abs(X(j,i)) |
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*> |
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*> The array is indexed by the right-hand side i (on which the |
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*> componentwise relative error depends), and the type of error |
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*> information as described below. There currently are up to three |
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*> pieces of information returned for each right-hand side. If |
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*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then |
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*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most |
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*> the first (:,N_ERR_BNDS) entries are returned. |
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*> |
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*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith |
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*> right-hand side. |
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*> |
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*> The second index in ERR_BNDS_COMP(:,err) contains the following |
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*> three fields: |
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*> err = 1 "Trust/don't trust" boolean. Trust the answer if the |
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*> reciprocal condition number is less than the threshold |
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*> sqrt(n) * dlamch('Epsilon'). |
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*> |
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*> err = 2 "Guaranteed" error bound: The estimated forward error, |
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*> almost certainly within a factor of 10 of the true error |
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*> so long as the next entry is greater than the threshold |
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*> sqrt(n) * dlamch('Epsilon'). This error bound should only |
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*> be trusted if the previous boolean is true. |
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*> |
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*> err = 3 Reciprocal condition number: Estimated componentwise |
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*> reciprocal condition number. Compared with the threshold |
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*> sqrt(n) * dlamch('Epsilon') to determine if the error |
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*> estimate is "guaranteed". These reciprocal condition |
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*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some |
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*> appropriately scaled matrix Z. |
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*> Let Z = S*(A*diag(x)), where x is the solution for the |
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*> current right-hand side and S scales each row of |
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*> A*diag(x) by a power of the radix so all absolute row |
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*> sums of Z are approximately 1. |
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*> |
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*> See Lapack Working Note 165 for further details and extra |
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*> cautions. |
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*> \endverbatim |
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*> |
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*> \param[in] NPARAMS |
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*> \verbatim |
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*> NPARAMS is INTEGER |
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*> Specifies the number of parameters set in PARAMS. If .LE. 0, the |
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*> PARAMS array is never referenced and default values are used. |
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*> \endverbatim |
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*> |
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*> \param[in,out] PARAMS |
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*> \verbatim |
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*> PARAMS is DOUBLE PRECISION array, dimension (NPARAMS) |
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*> Specifies algorithm parameters. If an entry is .LT. 0.0, then |
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*> that entry will be filled with default value used for that |
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*> parameter. Only positions up to NPARAMS are accessed; defaults |
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*> are used for higher-numbered parameters. |
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*> |
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*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative |
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*> refinement or not. |
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*> Default: 1.0D+0 |
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*> = 0.0 : No refinement is performed, and no error bounds are |
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*> computed. |
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*> = 1.0 : Use the double-precision refinement algorithm, |
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*> possibly with doubled-single computations if the |
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*> compilation environment does not support DOUBLE |
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*> PRECISION. |
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*> (other values are reserved for future use) |
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*> |
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*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual |
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*> computations allowed for refinement. |
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*> Default: 10 |
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*> Aggressive: Set to 100 to permit convergence using approximate |
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*> factorizations or factorizations other than LU. If |
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*> the factorization uses a technique other than |
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*> Gaussian elimination, the guarantees in |
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*> err_bnds_norm and err_bnds_comp may no longer be |
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*> trustworthy. |
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*> |
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*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code |
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*> will attempt to find a solution with small componentwise |
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*> relative error in the double-precision algorithm. Positive |
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*> is true, 0.0 is false. |
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*> Default: 1.0 (attempt componentwise convergence) |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (4*N) |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: Successful exit. The solution to every right-hand side is |
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*> guaranteed. |
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*> < 0: If INFO = -i, the i-th argument had an illegal value |
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*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization |
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*> has been completed, but the factor U is exactly singular, so |
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*> the solution and error bounds could not be computed. RCOND = 0 |
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*> is returned. |
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*> = N+J: The solution corresponding to the Jth right-hand side is |
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*> not guaranteed. The solutions corresponding to other right- |
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*> hand sides K with K > J may not be guaranteed as well, but |
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*> only the first such right-hand side is reported. If a small |
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*> componentwise error is not requested (PARAMS(3) = 0.0) then |
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*> the Jth right-hand side is the first with a normwise error |
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*> bound that is not guaranteed (the smallest J such |
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*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) |
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*> the Jth right-hand side is the first with either a normwise or |
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*> componentwise error bound that is not guaranteed (the smallest |
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*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or |
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*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of |
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*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information |
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*> about all of the right-hand sides check ERR_BNDS_NORM or |
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*> ERR_BNDS_COMP. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date April 2012 |
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* |
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*> \ingroup doublePOcomputational |
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* |
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* ===================================================================== |
SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, |
SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, |
$ LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, |
$ LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, |
$ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, |
$ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, |
$ WORK, IWORK, INFO ) |
$ WORK, IWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.2) -- |
* -- LAPACK computational routine (version 3.7.0) -- |
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Jason Riedy of Univ. of California Berkeley. -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- June 2010 -- |
* April 2012 |
* |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley and NAG Ltd. -- |
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* |
* |
IMPLICIT NONE |
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* .. |
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* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO, EQUED |
CHARACTER UPLO, EQUED |
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, |
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, |
Line 28
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Line 414
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$ ERR_BNDS_COMP( NRHS, * ) |
$ ERR_BNDS_COMP( NRHS, * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ================================================================== |
* ======= |
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* |
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* DPORFSX improves the computed solution to a system of linear |
|
* equations when the coefficient matrix is symmetric positive |
|
* definite, 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 and S |
|
* 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. |
|
* |
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* UPLO (input) CHARACTER*1 |
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* = 'U': Upper triangle of A is stored; |
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* = 'L': Lower triangle of A is stored. |
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* |
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* EQUED (input) CHARACTER*1 |
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* Specifies the form of equilibration that was done to A |
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* before calling this routine. This is needed to compute |
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* the solution and error bounds correctly. |
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* = 'N': No equilibration |
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* = 'Y': Both row and column equilibration, i.e., A has been |
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* replaced by diag(S) * A * diag(S). |
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* The right hand side B has been changed accordingly. |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* NRHS (input) INTEGER |
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* The number of right hand sides, i.e., the number of columns |
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* of the matrices B and X. NRHS >= 0. |
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* |
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* A (input) DOUBLE PRECISION array, dimension (LDA,N) |
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* The symmetric matrix A. If UPLO = 'U', the leading N-by-N |
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* upper triangular part of A contains the upper triangular part |
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* of the matrix A, and the strictly lower triangular part of A |
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* is not referenced. If UPLO = 'L', the leading N-by-N lower |
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* triangular part of A contains the lower triangular part of |
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* the matrix A, and the strictly upper triangular part of A is |
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* not referenced. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,N). |
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* |
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* AF (input) DOUBLE PRECISION array, dimension (LDAF,N) |
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* The triangular factor U or L from the Cholesky factorization |
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* A = U**T*U or A = L*L**T, as computed by DPOTRF. |
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* |
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* LDAF (input) INTEGER |
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* The leading dimension of the array AF. LDAF >= max(1,N). |
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* |
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* S (input or output) DOUBLE PRECISION array, dimension (N) |
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* The row scale factors for A. If EQUED = 'Y', A is multiplied on |
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* the left and right by diag(S). S is an input argument if FACT = |
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* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED |
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* = 'Y', each element of S must be positive. If S is output, each |
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* element of S is a power of the radix. If S is input, each element |
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* of S 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. |
|
* |
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* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) |
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* The right hand side matrix B. |
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* |
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* LDB (input) INTEGER |
|
* The leading dimension of the array B. LDB >= max(1,N). |
|
* |
|
* X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) |
|
* On entry, the solution matrix X, as computed by DGETRS. |
|
* On exit, the improved solution matrix X. |
|
* |
|
* LDX (input) INTEGER |
|
* The leading dimension of the array X. LDX >= max(1,N). |
|
* |
|
* RCOND (output) 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 (output) 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 (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) 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) * dlamch('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) * 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 "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) 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) * dlamch('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) * 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 "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) DOUBLE PRECISION 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: 1.0D+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 (workspace) DOUBLE PRECISION 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. |
|
* |
|
* ================================================================== |
|
* |
* |
* .. Parameters .. |
* .. Parameters .. |
DOUBLE PRECISION ZERO, ONE |
DOUBLE PRECISION ZERO, ONE |
Line 332
|
Line 455
|
INTRINSIC MAX, SQRT |
INTRINSIC MAX, SQRT |
* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
EXTERNAL LSAME, BLAS_FPINFO_X, ILATRANS, ILAPREC |
EXTERNAL LSAME, ILAPREC |
EXTERNAL DLAMCH, DLANSY, DLA_PORCOND |
EXTERNAL DLAMCH, DLANSY, DLA_PORCOND |
DOUBLE PRECISION DLAMCH, DLANSY, DLA_PORCOND |
DOUBLE PRECISION DLAMCH, DLANSY, DLA_PORCOND |
LOGICAL LSAME |
LOGICAL LSAME |
INTEGER BLAS_FPINFO_X |
INTEGER ILAPREC |
INTEGER ILATRANS, ILAPREC |
|
* .. |
* .. |
* .. Executable Statements .. |
* .. Executable Statements .. |
* |
* |