version 1.3, 2010/08/13 21:03:59
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version 1.9, 2012/12/14 12:30:27
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*> \brief \b DSYSVXX |
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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 DSYSVXX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysvxx.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/dsysvxx.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/dsysvxx.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 DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, |
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* EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, |
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* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, |
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* NPARAMS, PARAMS, WORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER EQUED, FACT, UPLO |
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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, RPVGRW |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IPIV( * ), 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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*> DSYSVXX uses the diagonal pivoting factorization to compute the |
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*> solution to a double precision system of linear equations A * X = B, where A |
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*> is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. |
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*> |
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*> If requested, both normwise and maximum componentwise error bounds |
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*> are returned. DSYSVXX will return a solution with a tiny |
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*> guaranteed error (O(eps) where eps is the working machine |
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*> precision) unless the matrix is very ill-conditioned, in which |
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*> case a warning is returned. Relevant condition numbers also are |
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*> calculated and returned. |
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*> |
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*> DSYSVXX accepts user-provided factorizations and equilibration |
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*> factors; see the definitions of the FACT and EQUED options. |
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*> Solving with refinement and using a factorization from a previous |
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*> DSYSVXX call will also produce a solution with either O(eps) |
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*> errors or warnings, but we cannot make that claim for general |
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*> user-provided factorizations and equilibration factors if they |
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*> differ from what DSYSVXX would itself produce. |
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*> \endverbatim |
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* |
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*> \par Description: |
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* ================= |
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*> |
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*> \verbatim |
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*> |
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*> The following steps are performed: |
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*> |
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*> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate |
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*> the system: |
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*> |
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*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B |
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*> |
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*> Whether or not the system will be equilibrated depends on the |
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*> scaling of the matrix A, but if equilibration is used, A is |
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*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. |
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*> |
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*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor |
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*> the matrix A (after equilibration if FACT = 'E') as |
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*> |
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*> A = U * D * U**T, if UPLO = 'U', or |
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*> A = L * D * L**T, if UPLO = 'L', |
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*> |
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*> where U (or L) is a product of permutation and unit upper (lower) |
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*> triangular matrices, and D is symmetric and block diagonal with |
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*> 1-by-1 and 2-by-2 diagonal blocks. |
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*> |
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*> 3. If some D(i,i)=0, so that D is exactly singular, then the |
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*> routine returns with INFO = i. Otherwise, the factored form of A |
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*> is used to estimate the condition number of the matrix A (see |
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*> argument RCOND). If the reciprocal of the condition number is |
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*> less than machine precision, the routine still goes on to solve |
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*> for X and compute error bounds as described below. |
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*> |
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*> 4. The system of equations is solved for X using the factored form |
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*> of A. |
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*> |
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*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), |
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*> the routine will use iterative refinement to try to get a small |
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*> error and error bounds. Refinement calculates the residual to at |
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*> least twice the working precision. |
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*> |
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*> 6. If equilibration was used, the matrix X is premultiplied by |
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*> diag(R) so that it solves the original system before |
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*> equilibration. |
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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] FACT |
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*> \verbatim |
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*> FACT is CHARACTER*1 |
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*> Specifies whether or not the factored form of the matrix A is |
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*> supplied on entry, and if not, whether the matrix A should be |
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*> equilibrated before it is factored. |
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*> = 'F': On entry, AF and IPIV contain the factored form of A. |
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*> If EQUED is not 'N', the matrix A has been |
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*> equilibrated with scaling factors given by S. |
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*> A, AF, and IPIV are not modified. |
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*> = 'N': The matrix A will be copied to AF and factored. |
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*> = 'E': The matrix A will be equilibrated if necessary, then |
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*> copied to AF and factored. |
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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] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of linear equations, i.e., the order of the |
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*> 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,out] 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 |
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*> part of the matrix A, and the strictly lower triangular |
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*> part of A is not referenced. If UPLO = 'L', the leading |
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*> N-by-N lower triangular part of A contains the lower |
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*> triangular part of the matrix A, and the strictly upper |
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*> triangular part of A is not referenced. |
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*> |
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*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by |
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*> diag(S)*A*diag(S). |
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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,out] AF |
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*> \verbatim |
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*> AF is DOUBLE PRECISION array, dimension (LDAF,N) |
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*> If FACT = 'F', then AF is an input argument and on entry |
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*> contains the block diagonal matrix D and the multipliers |
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*> used to obtain the factor U or L from the factorization A = |
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*> U*D*U**T or A = L*D*L**T as computed by DSYTRF. |
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*> |
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*> If FACT = 'N', then AF is an output argument and on exit |
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*> returns the block diagonal matrix D and the multipliers |
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*> used to obtain the factor U or L from the factorization A = |
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*> U*D*U**T or A = L*D*L**T. |
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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] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> If FACT = 'F', then IPIV is an input argument and on entry |
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*> contains details of the interchanges and the block |
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*> structure of D, as determined by DSYTRF. If IPIV(k) > 0, |
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*> then rows and columns k and IPIV(k) were interchanged and |
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*> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and |
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*> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and |
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*> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 |
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*> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, |
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*> then rows and columns k+1 and -IPIV(k) were interchanged |
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*> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. |
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*> |
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*> If FACT = 'N', then IPIV is an output argument and on exit |
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*> contains details of the interchanges and the block |
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*> structure of D, as determined by DSYTRF. |
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*> \endverbatim |
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*> |
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*> \param[in,out] 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. |
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*> = 'N': No equilibration (always true if FACT = 'N'). |
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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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*> EQUED is an input argument if FACT = 'F'; otherwise, it is an |
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*> output argument. |
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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 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,out] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) |
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*> On entry, the N-by-NRHS right hand side matrix B. |
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*> On exit, |
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*> if EQUED = 'N', B is not modified; |
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*> if EQUED = 'Y', B is overwritten by diag(S)*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[out] X |
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*> \verbatim |
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) |
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*> If INFO = 0, the N-by-NRHS solution matrix X to the original |
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*> system of equations. Note that A and B are modified on exit if |
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*> EQUED .ne. 'N', and the solution to the equilibrated system is |
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*> inv(diag(S))*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] RPVGRW |
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*> \verbatim |
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*> RPVGRW is DOUBLE PRECISION |
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*> Reciprocal pivot growth. On exit, this contains the reciprocal |
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*> pivot growth factor norm(A)/norm(U). The "max absolute element" |
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*> norm is used. If this is much less than 1, then the stability of |
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*> the LU factorization of the (equilibrated) matrix A could be poor. |
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*> This also means that the solution X, estimated condition numbers, |
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*> and error bounds could be unreliable. If factorization fails with |
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*> 0<INFO<=N, then this contains the reciprocal pivot growth factor |
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*> for the leading INFO columns of A. |
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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 extra-precise refinement algorithm. |
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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 September 2012 |
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* |
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*> \ingroup doubleSYdriver |
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* |
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* ===================================================================== |
SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, |
SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, |
$ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, |
$ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, |
$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, |
$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, |
$ NPARAMS, PARAMS, WORK, IWORK, INFO ) |
$ NPARAMS, PARAMS, WORK, IWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.2) -- |
* -- LAPACK driver routine (version 3.4.2) -- |
* -- 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 -- |
* September 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 EQUED, FACT, UPLO |
CHARACTER EQUED, FACT, UPLO |
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 525
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$ ERR_BNDS_COMP( NRHS, * ) |
$ ERR_BNDS_COMP( NRHS, * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ================================================================== |
* ======= |
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* |
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* DSYSVXX uses the diagonal pivoting factorization to compute the |
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* solution to a double precision system of linear equations A * X = B, where A |
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* is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. |
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* |
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* If requested, both normwise and maximum componentwise error bounds |
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* are returned. DSYSVXX will return a solution with a tiny |
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* guaranteed error (O(eps) where eps is the working machine |
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* precision) unless the matrix is very ill-conditioned, in which |
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* case a warning is returned. Relevant condition numbers also are |
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* calculated and returned. |
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* |
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* DSYSVXX accepts user-provided factorizations and equilibration |
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* factors; see the definitions of the FACT and EQUED options. |
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* Solving with refinement and using a factorization from a previous |
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* DSYSVXX call will also produce a solution with either O(eps) |
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* errors or warnings, but we cannot make that claim for general |
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* user-provided factorizations and equilibration factors if they |
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* differ from what DSYSVXX would itself produce. |
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* |
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* Description |
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* =========== |
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* |
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* The following steps are performed: |
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* |
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* 1. If FACT = 'E', double precision scaling factors are computed to equilibrate |
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* the system: |
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* |
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* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B |
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* |
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* Whether or not the system will be equilibrated depends on the |
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* scaling of the matrix A, but if equilibration is used, A is |
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* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. |
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* |
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* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor |
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* the matrix A (after equilibration if FACT = 'E') as |
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* |
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* A = U * D * U**T, if UPLO = 'U', or |
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* A = L * D * L**T, if UPLO = 'L', |
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* |
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* where U (or L) is a product of permutation and unit upper (lower) |
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* triangular matrices, and D is symmetric and block diagonal with |
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* 1-by-1 and 2-by-2 diagonal blocks. |
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* |
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* 3. If some D(i,i)=0, so that D is exactly singular, then the |
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* routine returns with INFO = i. Otherwise, the factored form of A |
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* is used to estimate the condition number of the matrix A (see |
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* argument RCOND). If the reciprocal of the condition number is |
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* less than machine precision, the routine still goes on to solve |
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* for X and compute error bounds as described below. |
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* |
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* 4. The system of equations is solved for X using the factored form |
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* of A. |
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* |
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* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), |
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* the routine will use iterative refinement to try to get a small |
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* error and error bounds. Refinement calculates the residual to at |
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* least twice the working precision. |
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* |
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* 6. If equilibration was used, the matrix X is premultiplied by |
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* diag(R) so that it solves the original system before |
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* equilibration. |
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* |
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* Arguments |
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* ========= |
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* |
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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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* |
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* FACT (input) CHARACTER*1 |
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* Specifies whether or not the factored form of the matrix A is |
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* supplied on entry, and if not, whether the matrix A should be |
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* equilibrated before it is factored. |
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* = 'F': On entry, AF and IPIV contain the factored form of A. |
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* If EQUED is not 'N', the matrix A has been |
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* equilibrated with scaling factors given by S. |
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* A, AF, and IPIV are not modified. |
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* = 'N': The matrix A will be copied to AF and factored. |
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* = 'E': The matrix A will be equilibrated if necessary, then |
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* copied to AF and factored. |
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* |
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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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* N (input) INTEGER |
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* The number of linear equations, i.e., the order of the |
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* 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/output) 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 |
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* part of the matrix A, and the strictly lower triangular |
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* part of A is not referenced. If UPLO = 'L', the leading |
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* N-by-N lower triangular part of A contains the lower |
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* triangular part of the matrix A, and the strictly upper |
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* triangular part of A is not referenced. |
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* |
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* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by |
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* diag(S)*A*diag(S). |
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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 or output) DOUBLE PRECISION array, dimension (LDAF,N) |
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* If FACT = 'F', then AF is an input argument and on entry |
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* contains the block diagonal matrix D and the multipliers |
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* used to obtain the factor U or L from the factorization A = |
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* U*D*U**T or A = L*D*L**T as computed by DSYTRF. |
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* |
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* If FACT = 'N', then AF is an output argument and on exit |
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* returns the block diagonal matrix D and the multipliers |
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* used to obtain the factor U or L from the factorization A = |
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* U*D*U**T or A = L*D*L**T. |
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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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* IPIV (input or output) INTEGER array, dimension (N) |
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* If FACT = 'F', then IPIV is an input argument and on entry |
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* contains details of the interchanges and the block |
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* structure of D, as determined by DSYTRF. If IPIV(k) > 0, |
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* then rows and columns k and IPIV(k) were interchanged and |
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* D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and |
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* IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and |
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* -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 |
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* diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, |
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* then rows and columns k+1 and -IPIV(k) were interchanged |
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* and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. |
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* |
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* If FACT = 'N', then IPIV is an output argument and on exit |
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* contains details of the interchanges and the block |
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* structure of D, as determined by DSYTRF. |
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* |
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* EQUED (input or output) CHARACTER*1 |
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* Specifies the form of equilibration that was done. |
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* = 'N': No equilibration (always true if FACT = 'N'). |
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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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* EQUED is an input argument if FACT = 'F'; otherwise, it is an |
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* output argument. |
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* |
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* S (input or output) DOUBLE PRECISION array, dimension (N) |
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* The 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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* |
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* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) |
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* On entry, the N-by-NRHS right hand side matrix B. |
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* On exit, |
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* if EQUED = 'N', B is not modified; |
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* if EQUED = 'Y', B is overwritten by diag(S)*B; |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,N). |
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* |
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* X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) |
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* If INFO = 0, the N-by-NRHS solution matrix X to the original |
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* system of equations. Note that A and B are modified on exit if |
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* EQUED .ne. 'N', and the solution to the equilibrated system is |
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* inv(diag(S))*X. |
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* |
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* LDX (input) INTEGER |
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* The leading dimension of the array X. LDX >= max(1,N). |
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* |
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* RCOND (output) 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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* |
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* RPVGRW (output) DOUBLE PRECISION |
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* Reciprocal pivot growth. On exit, this contains the reciprocal |
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* pivot growth factor norm(A)/norm(U). The "max absolute element" |
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* norm is used. If this is much less than 1, then the stability of |
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* the LU factorization of the (equilibrated) matrix A could be poor. |
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* This also means that the solution X, estimated condition numbers, |
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* and error bounds could be unreliable. If factorization fails with |
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* 0<INFO<=N, then this contains the reciprocal pivot growth factor |
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* for the leading INFO columns of A. |
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* |
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* BERR (output) 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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* |
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* N_ERR_BNDS (input) 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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* |
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* ERR_BNDS_NORM (output) 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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* |
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* ERR_BNDS_COMP (output) 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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* |
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* NPARAMS (input) 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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* |
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* PARAMS (input / output) 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 extra-precise refinement algorithm. |
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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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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) |
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* |
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* IWORK (workspace) INTEGER array, dimension (N) |
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* |
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* INFO (output) 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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* |
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* ================================================================== |
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* |
* |
* .. Parameters .. |
* .. Parameters .. |
DOUBLE PRECISION ZERO, ONE |
DOUBLE PRECISION ZERO, ONE |