version 1.4, 2010/12/21 13:48:06
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version 1.12, 2016/08/27 15:27:11
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*> \brief <b> DSPOSV computes the solution to system of linear equations A * X = B for PO 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 DSPOSV + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsposv.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/dsposv.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/dsposv.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 DSPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, |
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* SWORK, ITER, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER UPLO |
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* INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS |
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* .. |
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* .. Array Arguments .. |
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* REAL SWORK( * ) |
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* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), |
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* $ X( LDX, * ) |
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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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*> DSPOSV computes the solution to a real system of linear equations |
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*> A * X = B, |
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*> where A is an N-by-N symmetric positive definite matrix and X and B |
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*> are N-by-NRHS matrices. |
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*> |
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*> DSPOSV first attempts to factorize the matrix in SINGLE PRECISION |
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*> and use this factorization within an iterative refinement procedure |
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*> to produce a solution with DOUBLE PRECISION normwise backward error |
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*> quality (see below). If the approach fails the method switches to a |
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*> DOUBLE PRECISION factorization and solve. |
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*> |
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*> The iterative refinement is not going to be a winning strategy if |
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*> the ratio SINGLE PRECISION performance over DOUBLE PRECISION |
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*> performance is too small. A reasonable strategy should take the |
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*> number of right-hand sides and the size of the matrix into account. |
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*> This might be done with a call to ILAENV in the future. Up to now, we |
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*> always try iterative refinement. |
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*> |
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*> The iterative refinement process is stopped if |
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*> ITER > ITERMAX |
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*> or for all the RHS we have: |
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*> RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX |
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*> where |
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*> o ITER is the number of the current iteration in the iterative |
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*> refinement process |
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*> o RNRM is the infinity-norm of the residual |
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*> o XNRM is the infinity-norm of the solution |
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*> o ANRM is the infinity-operator-norm of the matrix A |
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*> o EPS is the machine epsilon returned by DLAMCH('Epsilon') |
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*> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 |
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*> respectively. |
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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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*> \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 matrix B. 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, |
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*> dimension (LDA,N) |
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*> On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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*> N-by-N upper triangular part of A contains the upper |
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*> triangular part of the matrix A, and the strictly lower |
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*> triangular part of A is not referenced. If UPLO = 'L', the |
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*> leading 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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*> On exit, if iterative refinement has been successfully used |
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*> (INFO.EQ.0 and ITER.GE.0, see description below), then A is |
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*> unchanged, if double precision factorization has been used |
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*> (INFO.EQ.0 and ITER.LT.0, see description below), then the |
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*> array A contains the factor U or L from the Cholesky |
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*> factorization A = U**T*U or A = L*L**T. |
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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] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) |
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*> The N-by-NRHS right hand side matrix B. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[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. |
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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] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (N,NRHS) |
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*> This array is used to hold the residual vectors. |
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*> \endverbatim |
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*> |
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*> \param[out] SWORK |
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*> \verbatim |
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*> SWORK is REAL array, dimension (N*(N+NRHS)) |
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*> This array is used to use the single precision matrix and the |
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*> right-hand sides or solutions in single precision. |
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*> \endverbatim |
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*> |
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*> \param[out] ITER |
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*> \verbatim |
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*> ITER is INTEGER |
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*> < 0: iterative refinement has failed, double precision |
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*> factorization has been performed |
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*> -1 : the routine fell back to full precision for |
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*> implementation- or machine-specific reasons |
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*> -2 : narrowing the precision induced an overflow, |
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*> the routine fell back to full precision |
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*> -3 : failure of SPOTRF |
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*> -31: stop the iterative refinement after the 30th |
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*> iterations |
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*> > 0: iterative refinement has been successfully used. |
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*> Returns the number of iterations |
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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 |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> > 0: if INFO = i, the leading minor of order i of (DOUBLE |
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*> PRECISION) A is not positive definite, so the |
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*> factorization could not be completed, and the solution |
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*> has not been computed. |
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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 June 2016 |
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* |
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*> \ingroup doublePOsolve |
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* |
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* ===================================================================== |
SUBROUTINE DSPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, |
SUBROUTINE DSPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, |
+ SWORK, ITER, INFO ) |
$ SWORK, ITER, INFO ) |
* |
* |
* -- LAPACK PROTOTYPE driver routine (version 3.3.0) -- |
* -- LAPACK driver routine (version 3.6.1) -- |
* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* November 2010 |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* June 2016 |
* |
* |
* .. |
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* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS |
INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS |
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* .. Array Arguments .. |
* .. Array Arguments .. |
REAL SWORK( * ) |
REAL SWORK( * ) |
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), |
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ), |
+ X( LDX, * ) |
$ X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ===================================================================== |
* ======= |
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* |
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* DSPOSV computes the solution to a real system of linear equations |
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* A * X = B, |
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* where A is an N-by-N symmetric positive definite matrix and X and B |
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* are N-by-NRHS matrices. |
|
* |
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* DSPOSV first attempts to factorize the matrix in SINGLE PRECISION |
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* and use this factorization within an iterative refinement procedure |
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* to produce a solution with DOUBLE PRECISION normwise backward error |
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* quality (see below). If the approach fails the method switches to a |
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* DOUBLE PRECISION factorization and solve. |
|
* |
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* The iterative refinement is not going to be a winning strategy if |
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* the ratio SINGLE PRECISION performance over DOUBLE PRECISION |
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* performance is too small. A reasonable strategy should take the |
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* number of right-hand sides and the size of the matrix into account. |
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* This might be done with a call to ILAENV in the future. Up to now, we |
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* always try iterative refinement. |
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* |
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* The iterative refinement process is stopped if |
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* ITER > ITERMAX |
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* or for all the RHS we have: |
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* RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX |
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* where |
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* o ITER is the number of the current iteration in the iterative |
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* refinement process |
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* o RNRM is the infinity-norm of the residual |
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* o XNRM is the infinity-norm of the solution |
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* o ANRM is the infinity-operator-norm of the matrix A |
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* o EPS is the machine epsilon returned by DLAMCH('Epsilon') |
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* The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 |
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* respectively. |
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* |
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* Arguments |
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* ========= |
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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 matrix B. NRHS >= 0. |
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* |
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* A (input/output) DOUBLE PRECISION array, |
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* dimension (LDA,N) |
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* On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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* N-by-N upper triangular part of A contains the upper |
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* triangular part of the matrix A, and the strictly lower |
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* triangular part of A is not referenced. If UPLO = 'L', the |
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* leading 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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* On exit, if iterative refinement has been successfully used |
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* (INFO.EQ.0 and ITER.GE.0, see description below), then A is |
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* unchanged, if double precision factorization has been used |
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* (INFO.EQ.0 and ITER.LT.0, see description below), then the |
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* array A contains the factor U or L from the Cholesky |
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* factorization A = U**T*U or A = L*L**T. |
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* |
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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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* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) |
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* The N-by-NRHS right hand side matrix 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. |
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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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* WORK (workspace) DOUBLE PRECISION array, dimension (N,NRHS) |
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* This array is used to hold the residual vectors. |
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* |
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* SWORK (workspace) REAL array, dimension (N*(N+NRHS)) |
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* This array is used to use the single precision matrix and the |
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* right-hand sides or solutions in single precision. |
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* |
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* ITER (output) INTEGER |
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* < 0: iterative refinement has failed, double precision |
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* factorization has been performed |
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* -1 : the routine fell back to full precision for |
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* implementation- or machine-specific reasons |
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* -2 : narrowing the precision induced an overflow, |
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* the routine fell back to full precision |
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* -3 : failure of SPOTRF |
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* -31: stop the iterative refinement after the 30th |
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* iterations |
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* > 0: iterative refinement has been sucessfully used. |
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* Returns the number of iterations |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* > 0: if INFO = i, the leading minor of order i of (DOUBLE |
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* PRECISION) A is not positive definite, so the |
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* factorization could not be completed, and the solution |
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* has not been computed. |
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* |
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* ========= |
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* |
* |
* .. Parameters .. |
* .. Parameters .. |
LOGICAL DOITREF |
LOGICAL DOITREF |
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* |
* |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DAXPY, DSYMM, DLACPY, DLAT2S, DLAG2S, SLAG2D, |
EXTERNAL DAXPY, DSYMM, DLACPY, DLAT2S, DLAG2S, SLAG2D, |
+ SPOTRF, SPOTRS, XERBLA |
$ SPOTRF, SPOTRS, XERBLA |
* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
INTEGER IDAMAX |
INTEGER IDAMAX |
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* Quick return if (N.EQ.0). |
* Quick return if (N.EQ.0). |
* |
* |
IF( N.EQ.0 ) |
IF( N.EQ.0 ) |
+ RETURN |
$ RETURN |
* |
* |
* Skip single precision iterative refinement if a priori slower |
* Skip single precision iterative refinement if a priori slower |
* than double precision factorization. |
* than double precision factorization. |
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* Solve the system SA*SX = SB. |
* Solve the system SA*SX = SB. |
* |
* |
CALL SPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N, |
CALL SPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N, |
+ INFO ) |
$ INFO ) |
* |
* |
* Convert SX back to double precision |
* Convert SX back to double precision |
* |
* |
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CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
* |
* |
CALL DSYMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE, |
CALL DSYMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE, |
+ WORK, N ) |
$ WORK, N ) |
* |
* |
* Check whether the NRHS normwise backward errors satisfy the |
* Check whether the NRHS normwise backward errors satisfy the |
* stopping criterion. If yes, set ITER=0 and return. |
* stopping criterion. If yes, set ITER=0 and return. |
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XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) |
XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) |
RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) |
RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) |
IF( RNRM.GT.XNRM*CTE ) |
IF( RNRM.GT.XNRM*CTE ) |
+ GO TO 10 |
$ GO TO 10 |
END DO |
END DO |
* |
* |
* If we are here, the NRHS normwise backward errors satisfy the |
* If we are here, the NRHS normwise backward errors satisfy the |
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* Solve the system SA*SX = SR. |
* Solve the system SA*SX = SR. |
* |
* |
CALL SPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N, |
CALL SPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N, |
+ INFO ) |
$ INFO ) |
* |
* |
* Convert SX back to double precision and update the current |
* Convert SX back to double precision and update the current |
* iterate. |
* iterate. |
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CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
* |
* |
CALL DSYMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE, |
CALL DSYMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE, |
+ WORK, N ) |
$ WORK, N ) |
* |
* |
* Check whether the NRHS normwise backward errors satisfy the |
* Check whether the NRHS normwise backward errors satisfy the |
* stopping criterion. If yes, set ITER=IITER>0 and return. |
* stopping criterion. If yes, set ITER=IITER>0 and return. |
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XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) |
XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) ) |
RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) |
RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) ) |
IF( RNRM.GT.XNRM*CTE ) |
IF( RNRM.GT.XNRM*CTE ) |
+ GO TO 20 |
$ GO TO 20 |
END DO |
END DO |
* |
* |
* If we are here, the NRHS normwise backward errors satisfy the |
* If we are here, the NRHS normwise backward errors satisfy the |
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CALL DPOTRF( UPLO, N, A, LDA, INFO ) |
CALL DPOTRF( UPLO, N, A, LDA, INFO ) |
* |
* |
IF( INFO.NE.0 ) |
IF( INFO.NE.0 ) |
+ RETURN |
$ RETURN |
* |
* |
CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX ) |
CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX ) |
CALL DPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO ) |
CALL DPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO ) |