version 1.4, 2010/12/21 13:53:42
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version 1.10, 2014/01/27 09:28:31
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*> \brief \b ZGBRFSX |
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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 ZGBRFSX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbrfsx.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/zgbrfsx.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/zgbrfsx.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 ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, |
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* LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, |
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* BERR, N_ERR_BNDS, ERR_BNDS_NORM, |
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* ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER TRANS, EQUED |
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* INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, |
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* $ NPARAMS, 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 IPIV( * ) |
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* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), |
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* $ X( LDX , * ),WORK( * ) |
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* DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ), |
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* $ ERR_BNDS_NORM( NRHS, * ), |
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* $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) |
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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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*> ZGBRFSX improves the computed solution to a system of linear |
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*> equations 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, R |
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*> and C below. In this case, the solution and error bounds returned |
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*> are 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] TRANS |
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*> \verbatim |
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*> TRANS is CHARACTER*1 |
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*> Specifies the form of the system of equations: |
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*> = 'N': A * X = B (No transpose) |
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*> = 'T': A**T * X = B (Transpose) |
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*> = 'C': A**H * X = B (Conjugate transpose = Transpose) |
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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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*> = 'R': Row equilibration, i.e., A has been premultiplied by |
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*> diag(R). |
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*> = 'C': Column equilibration, i.e., A has been postmultiplied |
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*> by diag(C). |
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*> = 'B': Both row and column equilibration, i.e., A has been |
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*> replaced by diag(R) * A * diag(C). |
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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] KL |
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*> \verbatim |
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*> KL is INTEGER |
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*> The number of subdiagonals within the band of A. KL >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KU |
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*> \verbatim |
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*> KU is INTEGER |
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*> The number of superdiagonals within the band of A. KU >= 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] AB |
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*> \verbatim |
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*> AB is COMPLEX*16 array, dimension (LDAB,N) |
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*> The original band matrix A, stored in rows 1 to KL+KU+1. |
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*> The j-th column of A is stored in the j-th column of the |
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*> array AB as follows: |
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*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). |
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*> \endverbatim |
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*> |
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*> \param[in] LDAB |
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*> \verbatim |
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*> LDAB is INTEGER |
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*> The leading dimension of the array AB. LDAB >= KL+KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in] AFB |
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*> \verbatim |
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*> AFB is COMPLEX*16 array, dimension (LDAFB,N) |
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*> Details of the LU factorization of the band matrix A, as |
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*> computed by DGBTRF. U is stored as an upper triangular band |
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*> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and |
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*> the multipliers used during the factorization are stored in |
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*> rows KL+KU+2 to 2*KL+KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in] LDAFB |
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*> \verbatim |
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*> LDAFB is INTEGER |
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*> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. |
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*> \endverbatim |
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*> |
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*> \param[in] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> The pivot indices from DGETRF; for 1<=i<=N, row i of the |
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*> matrix was interchanged with row IPIV(i). |
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*> \endverbatim |
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*> |
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*> \param[in,out] R |
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*> \verbatim |
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*> R is DOUBLE PRECISION array, dimension (N) |
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*> The row scale factors for A. If EQUED = 'R' or 'B', A is |
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*> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R |
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*> is not accessed. R is an input argument if FACT = 'F'; |
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*> otherwise, R is an output argument. If FACT = 'F' and |
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*> EQUED = 'R' or 'B', each element of R must be positive. |
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*> If R is output, each element of R is a power of the radix. |
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*> If R is input, each element of R should be a power of the radix |
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*> to ensure a reliable solution and error estimates. Scaling by |
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*> powers of the radix does not cause rounding errors unless the |
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*> result underflows or overflows. Rounding errors during scaling |
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*> lead to refining with a matrix that is not equivalent to the |
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*> input matrix, producing error estimates that may not be |
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*> reliable. |
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*> \endverbatim |
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*> |
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*> \param[in,out] C |
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*> \verbatim |
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*> C is DOUBLE PRECISION array, dimension (N) |
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*> The column scale factors for A. If EQUED = 'C' or 'B', A is |
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*> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C |
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*> is not accessed. C is an input argument if FACT = 'F'; |
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*> otherwise, C is an output argument. If FACT = 'F' and |
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*> EQUED = 'C' or 'B', each element of C must be positive. |
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*> If C is output, each element of C is a power of the radix. |
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*> If C is input, each element of C should be a power of the radix |
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*> to ensure a reliable solution and error estimates. Scaling by |
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*> powers of the radix does not cause rounding errors unless the |
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*> result underflows or overflows. Rounding errors during scaling |
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*> lead to refining with a matrix that is not equivalent to the |
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*> input matrix, producing error estimates that may not be |
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*> 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (2*N) |
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*> \endverbatim |
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*> |
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*> \param[out] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (2*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 complex16GBcomputational |
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* |
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* ===================================================================== |
SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, |
SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, |
$ LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, |
$ LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, |
$ BERR, N_ERR_BNDS, ERR_BNDS_NORM, |
$ BERR, N_ERR_BNDS, ERR_BNDS_NORM, |
$ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, |
$ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, |
$ INFO ) |
$ INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.2) -- |
* -- LAPACK computational routine (version 3.4.1) -- |
* -- 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 TRANS, EQUED |
CHARACTER TRANS, EQUED |
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, |
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, |
Line 29
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Line 460
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$ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) |
$ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ================================================================== |
* ======= |
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* |
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* ZGBRFSX improves the computed solution to a system of linear |
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* equations and provides error bounds and backward error estimates |
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* 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. |
|
* |
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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, R |
|
* and C below. In this case, the solution and error bounds returned |
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* are for the original unequilibrated system. |
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* |
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* 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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* TRANS (input) CHARACTER*1 |
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* Specifies the form of the system of equations: |
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* = 'N': A * X = B (No transpose) |
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* = 'T': A**T * X = B (Transpose) |
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* = 'C': A**H * X = B (Conjugate transpose = Transpose) |
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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 |
|
* 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. |
|
* |
|
* KL (input) INTEGER |
|
* The number of subdiagonals within the band of A. KL >= 0. |
|
* |
|
* KU (input) INTEGER |
|
* The number of superdiagonals within the band of A. KU >= 0. |
|
* |
|
* NRHS (input) INTEGER |
|
* The number of right hand sides, i.e., the number of columns |
|
* of the matrices B and X. NRHS >= 0. |
|
* |
|
* AB (input) DOUBLE PRECISION array, dimension (LDAB,N) |
|
* The original band matrix A, stored in rows 1 to KL+KU+1. |
|
* The j-th column of A is stored in the j-th column of the |
|
* array AB as follows: |
|
* AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). |
|
* |
|
* LDAB (input) INTEGER |
|
* The leading dimension of the array AB. LDAB >= KL+KU+1. |
|
* |
|
* AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) |
|
* Details of the LU factorization of the band matrix A, as |
|
* computed by DGBTRF. U is stored as an upper triangular band |
|
* matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and |
|
* the multipliers used during the factorization are stored in |
|
* rows KL+KU+2 to 2*KL+KU+1. |
|
* |
|
* LDAFB (input) INTEGER |
|
* The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. |
|
* |
|
* IPIV (input) INTEGER array, dimension (N) |
|
* The pivot indices from DGETRF; for 1<=i<=N, row i of the |
|
* matrix was interchanged with row IPIV(i). |
|
* |
|
* R (input or output) DOUBLE PRECISION array, dimension (N) |
|
* The row scale factors for A. If EQUED = 'R' or 'B', A is |
|
* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R |
|
* is not accessed. R is an input argument if FACT = 'F'; |
|
* otherwise, R is an output argument. If FACT = 'F' and |
|
* EQUED = 'R' or 'B', each element of R must be positive. |
|
* If R is output, each element of R is a power of the radix. |
|
* If R is input, 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 or output) DOUBLE PRECISION array, dimension (N) |
|
* The column scale factors for A. If EQUED = 'C' or 'B', A is |
|
* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C |
|
* is not accessed. C is an input argument if FACT = 'F'; |
|
* otherwise, C is an output argument. If FACT = 'F' and |
|
* EQUED = 'C' or 'B', each element of C must be positive. |
|
* If C is output, each element of C is a power of the radix. |
|
* 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 (input) DOUBLE PRECISION 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) 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) COMPLEX*16 array, dimension (2*N) |
|
* |
|
* RWORK (workspace) DOUBLE PRECISION array, dimension (2*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 |