version 1.3, 2010/08/13 21:04:03
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version 1.5, 2011/11/21 20:43:10
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*> \brief <b> ZGESVXX computes the solution to system of linear equations A * X = B for GE matrices</b> |
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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 ZGESVXX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgesvxx.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/zgesvxx.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/zgesvxx.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 ZGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, |
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* EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, |
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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 EQUED, FACT, TRANS |
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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( * ) |
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* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), 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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*> ZGESVXX uses the LU factorization to compute the solution to a |
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*> complex*16 system of linear equations A * X = B, where A is an |
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*> N-by-N 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. ZGESVXX 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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*> ZGESVXX 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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*> ZGESVXX 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 ZGESVXX 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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*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B |
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*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B |
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*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') |
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*> or diag(C)*B (if TRANS = 'T' or 'C'). |
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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 = P * L * U, |
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*> |
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*> where P is a permutation matrix, L is a unit lower triangular |
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*> matrix, and U is upper triangular. |
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*> |
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*> 3. If some U(i,i)=0, so that U 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 less |
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*> than machine precision, the routine still goes on to solve for X |
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*> 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so |
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*> that it solves the original system before 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 R and C. |
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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] 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) |
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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 COMPLEX*16 array, dimension (LDA,N) |
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*> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is |
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*> not 'N', then A must have been equilibrated by the scaling |
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*> factors in R and/or C. A is not modified if FACT = 'F' or |
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*> 'N', or if FACT = 'E' and EQUED = 'N' on exit. |
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*> |
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*> On exit, if EQUED .ne. 'N', A is scaled as follows: |
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*> EQUED = 'R': A := diag(R) * A |
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*> EQUED = 'C': A := A * diag(C) |
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*> EQUED = 'B': A := diag(R) * A * diag(C). |
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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 or output) COMPLEX*16 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 factors L and U from the factorization |
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*> A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then |
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*> AF is the factored form of the equilibrated matrix A. |
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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 factors L and U from the factorization A = P*L*U |
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*> of the original matrix A. |
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*> |
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*> If FACT = 'E', then AF is an output argument and on exit |
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*> returns the factors L and U from the factorization A = P*L*U |
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*> of the equilibrated matrix A (see the description of A for |
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*> the form of the equilibrated matrix). |
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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 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 the pivot indices from the factorization A = P*L*U |
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*> as computed by ZGETRF; row i of the matrix was interchanged |
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*> with row IPIV(i). |
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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 the pivot indices from the factorization A = P*L*U |
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*> of the original matrix A. |
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*> |
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*> If FACT = 'E', then IPIV is an output argument and on exit |
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*> contains the pivot indices from the factorization A = P*L*U |
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*> of the equilibrated matrix A. |
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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 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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*> = '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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*> 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] R |
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*> \verbatim |
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*> R is or output) 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 or output) 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,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by |
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*> diag(R)*B; |
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*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is |
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*> overwritten by diag(C)*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 COMPLEX*16 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 |
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*> if EQUED .ne. 'N', and the solution to the equilibrated system is |
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*> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or |
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*> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. |
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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. In ZGESVX, this quantity is |
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*> returned in WORK(1). |
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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 / 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 |
|
*> relative error in the double-precision algorithm. Positive |
|
*> is true, 0.0 is false. |
|
*> Default: 1.0 (attempt componentwise convergence) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] WORK |
|
*> \verbatim |
|
*> WORK is COMPLEX*16 array, dimension (2*N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] RWORK |
|
*> \verbatim |
|
*> RWORK is DOUBLE PRECISION array, dimension (2*N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] INFO |
|
*> \verbatim |
|
*> INFO is 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. |
|
*> \endverbatim |
|
* |
|
* Authors: |
|
* ======== |
|
* |
|
*> \author Univ. of Tennessee |
|
*> \author Univ. of California Berkeley |
|
*> \author Univ. of Colorado Denver |
|
*> \author NAG Ltd. |
|
* |
|
*> \date November 2011 |
|
* |
|
*> \ingroup complex16GEsolve |
|
* |
|
* ===================================================================== |
SUBROUTINE ZGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, |
SUBROUTINE ZGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, |
$ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, |
$ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW, |
$ 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 driver routine (version 3.2.1) -- |
* -- LAPACK driver routine (version 3.4.0) -- |
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Jason Riedy of Univ. of California Berkeley. -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- April 2009 -- |
* November 2011 |
* |
|
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
|
* -- Univ. of California Berkeley and NAG Ltd. -- |
|
* |
* |
IMPLICIT NONE |
|
* .. |
|
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER EQUED, FACT, TRANS |
CHARACTER EQUED, FACT, TRANS |
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, |
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, |
Line 29
|
Line 560
|
$ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) |
$ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ================================================================== |
* ======= |
|
* |
|
* ZGESVXX uses the LU factorization to compute the solution to a |
|
* complex*16 system of linear equations A * X = B, where A is an |
|
* N-by-N matrix and X and B are N-by-NRHS matrices. |
|
* |
|
* If requested, both normwise and maximum componentwise error bounds |
|
* are returned. ZGESVXX will return a solution with a tiny |
|
* guaranteed error (O(eps) where eps is the working machine |
|
* precision) unless the matrix is very ill-conditioned, in which |
|
* case a warning is returned. Relevant condition numbers also are |
|
* calculated and returned. |
|
* |
|
* ZGESVXX accepts user-provided factorizations and equilibration |
|
* factors; see the definitions of the FACT and EQUED options. |
|
* Solving with refinement and using a factorization from a previous |
|
* ZGESVXX call will also produce a solution with either O(eps) |
|
* errors or warnings, but we cannot make that claim for general |
|
* user-provided factorizations and equilibration factors if they |
|
* differ from what ZGESVXX would itself produce. |
|
* |
|
* Description |
|
* =========== |
|
* |
|
* The following steps are performed: |
|
* |
|
* 1. If FACT = 'E', double precision scaling factors are computed to equilibrate |
|
* the system: |
|
* |
|
* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B |
|
* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B |
|
* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B |
|
* |
|
* Whether or not the system will be equilibrated depends on the |
|
* scaling of the matrix A, but if equilibration is used, A is |
|
* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') |
|
* or diag(C)*B (if TRANS = 'T' or 'C'). |
|
* |
|
* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor |
|
* the matrix A (after equilibration if FACT = 'E') as |
|
* |
|
* A = P * L * U, |
|
* |
|
* where P is a permutation matrix, L is a unit lower triangular |
|
* matrix, and U is upper triangular. |
|
* |
|
* 3. If some U(i,i)=0, so that U is exactly singular, then the |
|
* routine returns with INFO = i. Otherwise, the factored form of A |
|
* is used to estimate the condition number of the matrix A (see |
|
* argument RCOND). If the reciprocal of the condition number is less |
|
* than machine precision, the routine still goes on to solve for X |
|
* and compute error bounds as described below. |
|
* |
|
* 4. The system of equations is solved for X using the factored form |
|
* of A. |
|
* |
|
* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), |
|
* the routine will use iterative refinement to try to get a small |
|
* error and error bounds. Refinement calculates the residual to at |
|
* least twice the working precision. |
|
* |
|
* 6. If equilibration was used, the matrix X is premultiplied by |
|
* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so |
|
* that it solves the original system before equilibration. |
|
* |
|
* 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. |
|
* |
|
* FACT (input) CHARACTER*1 |
|
* Specifies whether or not the factored form of the matrix A is |
|
* supplied on entry, and if not, whether the matrix A should be |
|
* equilibrated before it is factored. |
|
* = 'F': On entry, AF and IPIV contain the factored form of A. |
|
* If EQUED is not 'N', the matrix A has been |
|
* equilibrated with scaling factors given by R and C. |
|
* A, AF, and IPIV are not modified. |
|
* = 'N': The matrix A will be copied to AF and factored. |
|
* = 'E': The matrix A will be equilibrated if necessary, then |
|
* copied to AF and factored. |
|
* |
|
* TRANS (input) CHARACTER*1 |
|
* Specifies the form of the system of equations: |
|
* = 'N': A * X = B (No transpose) |
|
* = 'T': A**T * X = B (Transpose) |
|
* = 'C': A**H * X = B (Conjugate Transpose) |
|
* |
|
* N (input) INTEGER |
|
* The number of linear equations, i.e., the order of the |
|
* matrix A. N >= 0. |
|
* |
|
* NRHS (input) INTEGER |
|
* The number of right hand sides, i.e., the number of columns |
|
* of the matrices B and X. NRHS >= 0. |
|
* |
|
* A (input/output) COMPLEX*16 array, dimension (LDA,N) |
|
* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is |
|
* not 'N', then A must have been equilibrated by the scaling |
|
* factors in R and/or C. A is not modified if FACT = 'F' or |
|
* 'N', or if FACT = 'E' and EQUED = 'N' on exit. |
|
* |
|
* On exit, if EQUED .ne. 'N', A is scaled as follows: |
|
* EQUED = 'R': A := diag(R) * A |
|
* EQUED = 'C': A := A * diag(C) |
|
* EQUED = 'B': A := diag(R) * A * diag(C). |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
|
* AF (input or output) COMPLEX*16 array, dimension (LDAF,N) |
|
* If FACT = 'F', then AF is an input argument and on entry |
|
* contains the factors L and U from the factorization |
|
* A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then |
|
* AF is the factored form of the equilibrated matrix A. |
|
* |
|
* If FACT = 'N', then AF is an output argument and on exit |
|
* returns the factors L and U from the factorization A = P*L*U |
|
* of the original matrix A. |
|
* |
|
* If FACT = 'E', then AF is an output argument and on exit |
|
* returns the factors L and U from the factorization A = P*L*U |
|
* of the equilibrated matrix A (see the description of A for |
|
* the form of the equilibrated matrix). |
|
* |
|
* LDAF (input) INTEGER |
|
* The leading dimension of the array AF. LDAF >= max(1,N). |
|
* |
|
* IPIV (input or output) INTEGER array, dimension (N) |
|
* If FACT = 'F', then IPIV is an input argument and on entry |
|
* contains the pivot indices from the factorization A = P*L*U |
|
* as computed by ZGETRF; row i of the matrix was interchanged |
|
* with row IPIV(i). |
|
* |
|
* If FACT = 'N', then IPIV is an output argument and on exit |
|
* contains the pivot indices from the factorization A = P*L*U |
|
* of the original matrix A. |
|
* |
|
* If FACT = 'E', then IPIV is an output argument and on exit |
|
* contains the pivot indices from the factorization A = P*L*U |
|
* of the equilibrated matrix A. |
|
* |
|
* EQUED (input or output) CHARACTER*1 |
|
* Specifies the form of equilibration that was done. |
|
* = 'N': No equilibration (always true if FACT = 'N'). |
|
* = '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). |
|
* EQUED is an input argument if FACT = 'F'; otherwise, it is an |
|
* output argument. |
|
* |
|
* 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/output) COMPLEX*16 array, dimension (LDB,NRHS) |
|
* On entry, the N-by-NRHS right hand side matrix B. |
|
* On exit, |
|
* if EQUED = 'N', B is not modified; |
|
* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by |
|
* diag(R)*B; |
|
* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is |
|
* overwritten by diag(C)*B. |
|
* |
|
* LDB (input) INTEGER |
|
* The leading dimension of the array B. LDB >= max(1,N). |
|
* |
|
* X (output) COMPLEX*16 array, dimension (LDX,NRHS) |
|
* If INFO = 0, the N-by-NRHS solution matrix X to the original |
|
* system of equations. Note that A and B are modified on exit |
|
* if EQUED .ne. 'N', and the solution to the equilibrated system is |
|
* inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or |
|
* inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. |
|
* |
|
* 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. |
|
* |
|
* RPVGRW (output) DOUBLE PRECISION |
|
* Reciprocal pivot growth. On exit, this contains the reciprocal |
|
* pivot growth factor norm(A)/norm(U). The "max absolute element" |
|
* norm is used. If this is much less than 1, then the stability of |
|
* the LU factorization of the (equilibrated) matrix A could be poor. |
|
* This also means that the solution X, estimated condition numbers, |
|
* and error bounds could be unreliable. If factorization fails with |
|
* 0<INFO<=N, then this contains the reciprocal pivot growth factor |
|
* for the leading INFO columns of A. In ZGESVX, this quantity is |
|
* returned in WORK(1). |
|
* |
|
* 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 extra-precise refinement algorithm. |
|
* (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 |
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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 |