version 1.5, 2010/08/07 13:18:08
|
version 1.12, 2012/08/22 09:48:28
|
Line 1
|
Line 1
|
|
*> \brief <b> ZCGESV computes the solution to system of linear equations A * X = B for GE matrices</b> (mixed precision with iterative refinement) |
|
* |
|
* =========== DOCUMENTATION =========== |
|
* |
|
* Online html documentation available at |
|
* http://www.netlib.org/lapack/explore-html/ |
|
* |
|
*> \htmlonly |
|
*> Download ZCGESV + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zcgesv.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zcgesv.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zcgesv.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE ZCGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, |
|
* SWORK, RWORK, ITER, INFO ) |
|
* |
|
* .. Scalar Arguments .. |
|
* INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS |
|
* .. |
|
* .. Array Arguments .. |
|
* INTEGER IPIV( * ) |
|
* DOUBLE PRECISION RWORK( * ) |
|
* COMPLEX SWORK( * ) |
|
* COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ), |
|
* $ X( LDX, * ) |
|
* .. |
|
* |
|
* |
|
*> \par Purpose: |
|
* ============= |
|
*> |
|
*> \verbatim |
|
*> |
|
*> ZCGESV computes the solution to a complex system of linear equations |
|
*> A * X = B, |
|
*> where A is an N-by-N matrix and X and B are N-by-NRHS matrices. |
|
*> |
|
*> ZCGESV first attempts to factorize the matrix in COMPLEX and use this |
|
*> factorization within an iterative refinement procedure to produce a |
|
*> solution with COMPLEX*16 normwise backward error quality (see below). |
|
*> If the approach fails the method switches to a COMPLEX*16 |
|
*> factorization and solve. |
|
*> |
|
*> The iterative refinement is not going to be a winning strategy if |
|
*> the ratio COMPLEX performance over COMPLEX*16 performance is too |
|
*> small. A reasonable strategy should take the number of right-hand |
|
*> sides and the size of the matrix into account. This might be done |
|
*> with a call to ILAENV in the future. Up to now, we always try |
|
*> iterative refinement. |
|
*> |
|
*> The iterative refinement process is stopped if |
|
*> ITER > ITERMAX |
|
*> or for all the RHS we have: |
|
*> RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX |
|
*> where |
|
*> o ITER is the number of the current iteration in the iterative |
|
*> refinement process |
|
*> o RNRM is the infinity-norm of the residual |
|
*> o XNRM is the infinity-norm of the solution |
|
*> o ANRM is the infinity-operator-norm of the matrix A |
|
*> o EPS is the machine epsilon returned by DLAMCH('Epsilon') |
|
*> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 |
|
*> respectively. |
|
*> \endverbatim |
|
* |
|
* Arguments: |
|
* ========== |
|
* |
|
*> \param[in] N |
|
*> \verbatim |
|
*> N is INTEGER |
|
*> The number of linear equations, i.e., the order of the |
|
*> matrix A. N >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] NRHS |
|
*> \verbatim |
|
*> NRHS is INTEGER |
|
*> The number of right hand sides, i.e., the number of columns |
|
*> of the matrix B. NRHS >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] A |
|
*> \verbatim |
|
*> A is COMPLEX*16 array, |
|
*> dimension (LDA,N) |
|
*> On entry, the N-by-N coefficient matrix A. |
|
*> On exit, if iterative refinement has been successfully used |
|
*> (INFO.EQ.0 and ITER.GE.0, see description below), then A is |
|
*> unchanged, if double precision factorization has been used |
|
*> (INFO.EQ.0 and ITER.LT.0, see description below), then the |
|
*> array A contains the factors L and U from the factorization |
|
*> A = P*L*U; the unit diagonal elements of L are not stored. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDA |
|
*> \verbatim |
|
*> LDA is INTEGER |
|
*> The leading dimension of the array A. LDA >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[out] IPIV |
|
*> \verbatim |
|
*> IPIV is INTEGER array, dimension (N) |
|
*> The pivot indices that define the permutation matrix P; |
|
*> row i of the matrix was interchanged with row IPIV(i). |
|
*> Corresponds either to the single precision factorization |
|
*> (if INFO.EQ.0 and ITER.GE.0) or the double precision |
|
*> factorization (if INFO.EQ.0 and ITER.LT.0). |
|
*> \endverbatim |
|
*> |
|
*> \param[in] B |
|
*> \verbatim |
|
*> B is COMPLEX*16 array, dimension (LDB,NRHS) |
|
*> The N-by-NRHS right hand side matrix B. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDB |
|
*> \verbatim |
|
*> LDB is INTEGER |
|
*> The leading dimension of the array B. LDB >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[out] X |
|
*> \verbatim |
|
*> X is COMPLEX*16 array, dimension (LDX,NRHS) |
|
*> If INFO = 0, the N-by-NRHS solution matrix X. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDX |
|
*> \verbatim |
|
*> LDX is INTEGER |
|
*> The leading dimension of the array X. LDX >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[out] WORK |
|
*> \verbatim |
|
*> WORK is COMPLEX*16 array, dimension (N*NRHS) |
|
*> This array is used to hold the residual vectors. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] SWORK |
|
*> \verbatim |
|
*> SWORK is COMPLEX array, dimension (N*(N+NRHS)) |
|
*> This array is used to use the single precision matrix and the |
|
*> right-hand sides or solutions in single precision. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] RWORK |
|
*> \verbatim |
|
*> RWORK is DOUBLE PRECISION array, dimension (N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] ITER |
|
*> \verbatim |
|
*> ITER is INTEGER |
|
*> < 0: iterative refinement has failed, COMPLEX*16 |
|
*> factorization has been performed |
|
*> -1 : the routine fell back to full precision for |
|
*> implementation- or machine-specific reasons |
|
*> -2 : narrowing the precision induced an overflow, |
|
*> the routine fell back to full precision |
|
*> -3 : failure of CGETRF |
|
*> -31: stop the iterative refinement after the 30th |
|
*> iterations |
|
*> > 0: iterative refinement has been sucessfully used. |
|
*> Returns the number of iterations |
|
*> \endverbatim |
|
*> |
|
*> \param[out] INFO |
|
*> \verbatim |
|
*> INFO is INTEGER |
|
*> = 0: successful exit |
|
*> < 0: if INFO = -i, the i-th argument had an illegal value |
|
*> > 0: if INFO = i, U(i,i) computed in COMPLEX*16 is exactly |
|
*> zero. The factorization has been completed, but the |
|
*> factor U is exactly singular, so the solution |
|
*> could not be computed. |
|
*> \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 ZCGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, |
SUBROUTINE ZCGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK, |
+ SWORK, RWORK, ITER, INFO ) |
$ SWORK, RWORK, ITER, INFO ) |
* |
* |
* -- LAPACK PROTOTYPE driver routine (version 3.2.2) -- |
* -- LAPACK driver routine (version 3.4.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* January 2007 |
* November 2011 |
* |
* |
* .. |
|
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS |
INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS |
* .. |
* .. |
Line 15
|
Line 214
|
DOUBLE PRECISION RWORK( * ) |
DOUBLE PRECISION RWORK( * ) |
COMPLEX SWORK( * ) |
COMPLEX SWORK( * ) |
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ), |
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ), |
+ X( LDX, * ) |
$ X( LDX, * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ===================================================================== |
* ======= |
|
* |
|
* ZCGESV computes the solution to a complex system of linear equations |
|
* A * X = B, |
|
* where A is an N-by-N matrix and X and B are N-by-NRHS matrices. |
|
* |
|
* ZCGESV first attempts to factorize the matrix in COMPLEX and use this |
|
* factorization within an iterative refinement procedure to produce a |
|
* solution with COMPLEX*16 normwise backward error quality (see below). |
|
* If the approach fails the method switches to a COMPLEX*16 |
|
* factorization and solve. |
|
* |
|
* The iterative refinement is not going to be a winning strategy if |
|
* the ratio COMPLEX performance over COMPLEX*16 performance is too |
|
* small. A reasonable strategy should take the number of right-hand |
|
* sides and the size of the matrix into account. This might be done |
|
* with a call to ILAENV in the future. Up to now, we always try |
|
* iterative refinement. |
|
* |
|
* The iterative refinement process is stopped if |
|
* ITER > ITERMAX |
|
* or for all the RHS we have: |
|
* RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX |
|
* where |
|
* o ITER is the number of the current iteration in the iterative |
|
* refinement process |
|
* o RNRM is the infinity-norm of the residual |
|
* o XNRM is the infinity-norm of the solution |
|
* o ANRM is the infinity-operator-norm of the matrix A |
|
* o EPS is the machine epsilon returned by DLAMCH('Epsilon') |
|
* The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 |
|
* respectively. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* 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 matrix B. NRHS >= 0. |
|
* |
|
* A (input/output) COMPLEX*16 array, |
|
* dimension (LDA,N) |
|
* On entry, the N-by-N coefficient matrix A. |
|
* On exit, if iterative refinement has been successfully used |
|
* (INFO.EQ.0 and ITER.GE.0, see description below), then A is |
|
* unchanged, if double precision factorization has been used |
|
* (INFO.EQ.0 and ITER.LT.0, see description below), then the |
|
* array A contains the factors L and U from the factorization |
|
* A = P*L*U; the unit diagonal elements of L are not stored. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
|
* IPIV (output) INTEGER array, dimension (N) |
|
* The pivot indices that define the permutation matrix P; |
|
* row i of the matrix was interchanged with row IPIV(i). |
|
* Corresponds either to the single precision factorization |
|
* (if INFO.EQ.0 and ITER.GE.0) or the double precision |
|
* factorization (if INFO.EQ.0 and ITER.LT.0). |
|
* |
|
* B (input) COMPLEX*16 array, dimension (LDB,NRHS) |
|
* The N-by-NRHS right hand side matrix 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. |
|
* |
|
* LDX (input) INTEGER |
|
* The leading dimension of the array X. LDX >= max(1,N). |
|
* |
|
* WORK (workspace) COMPLEX*16 array, dimension (N*NRHS) |
|
* This array is used to hold the residual vectors. |
|
* |
|
* SWORK (workspace) COMPLEX array, dimension (N*(N+NRHS)) |
|
* This array is used to use the single precision matrix and the |
|
* right-hand sides or solutions in single precision. |
|
* |
|
* RWORK (workspace) DOUBLE PRECISION array, dimension (N) |
|
* |
|
* ITER (output) INTEGER |
|
* < 0: iterative refinement has failed, COMPLEX*16 |
|
* factorization has been performed |
|
* -1 : the routine fell back to full precision for |
|
* implementation- or machine-specific reasons |
|
* -2 : narrowing the precision induced an overflow, |
|
* the routine fell back to full precision |
|
* -3 : failure of CGETRF |
|
* -31: stop the iterative refinement after the 30th |
|
* iterations |
|
* > 0: iterative refinement has been sucessfully used. |
|
* Returns the number of iterations |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value |
|
* > 0: if INFO = i, U(i,i) computed in COMPLEX*16 is exactly |
|
* zero. The factorization has been completed, but the |
|
* factor U is exactly singular, so the solution |
|
* could not be computed. |
|
* |
|
* ========= |
|
* |
* |
* .. Parameters .. |
* .. Parameters .. |
LOGICAL DOITREF |
LOGICAL DOITREF |
Line 139
|
Line 231
|
* |
* |
COMPLEX*16 NEGONE, ONE |
COMPLEX*16 NEGONE, ONE |
PARAMETER ( NEGONE = ( -1.0D+00, 0.0D+00 ), |
PARAMETER ( NEGONE = ( -1.0D+00, 0.0D+00 ), |
+ ONE = ( 1.0D+00, 0.0D+00 ) ) |
$ ONE = ( 1.0D+00, 0.0D+00 ) ) |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |
INTEGER I, IITER, PTSA, PTSX |
INTEGER I, IITER, PTSA, PTSX |
Line 148
|
Line 240
|
* |
* |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL CGETRS, CGETRF, CLAG2Z, XERBLA, ZAXPY, ZGEMM, |
EXTERNAL CGETRS, CGETRF, CLAG2Z, XERBLA, ZAXPY, ZGEMM, |
+ ZLACPY, ZLAG2C |
$ ZLACPY, ZLAG2C |
* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
INTEGER IZAMAX |
INTEGER IZAMAX |
Line 190
|
Line 282
|
* 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. |
Line 243
|
Line 335
|
* Solve the system SA*SX = SB. |
* Solve the system SA*SX = SB. |
* |
* |
CALL CGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, |
CALL CGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, |
+ SWORK( PTSX ), N, INFO ) |
$ SWORK( PTSX ), N, INFO ) |
* |
* |
* Convert SX back to double precision |
* Convert SX back to double precision |
* |
* |
Line 254
|
Line 346
|
CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
* |
* |
CALL ZGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, A, |
CALL ZGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, A, |
+ LDA, X, LDX, ONE, WORK, N ) |
$ LDA, X, LDX, ONE, 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. |
Line 263
|
Line 355
|
XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) ) |
XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) ) |
RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) ) |
RNRM = CABS1( WORK( IZAMAX( 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 |
Line 289
|
Line 381
|
* Solve the system SA*SX = SR. |
* Solve the system SA*SX = SR. |
* |
* |
CALL CGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, |
CALL CGETRS( 'No transpose', N, NRHS, SWORK( PTSA ), N, IPIV, |
+ SWORK( PTSX ), N, INFO ) |
$ SWORK( PTSX ), N, INFO ) |
* |
* |
* Convert SX back to double precision and update the current |
* Convert SX back to double precision and update the current |
* iterate. |
* iterate. |
Line 305
|
Line 397
|
CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N ) |
* |
* |
CALL ZGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, |
CALL ZGEMM( 'No Transpose', 'No Transpose', N, NRHS, N, NEGONE, |
+ A, LDA, X, LDX, ONE, WORK, N ) |
$ A, LDA, X, LDX, ONE, 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. |
Line 314
|
Line 406
|
XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) ) |
XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) ) |
RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) ) |
RNRM = CABS1( WORK( IZAMAX( 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 |
Line 343
|
Line 435
|
CALL ZGETRF( N, N, A, LDA, IPIV, INFO ) |
CALL ZGETRF( N, N, A, LDA, IPIV, INFO ) |
* |
* |
IF( INFO.NE.0 ) |
IF( INFO.NE.0 ) |
+ RETURN |
$ RETURN |
* |
* |
CALL ZLACPY( 'All', N, NRHS, B, LDB, X, LDX ) |
CALL ZLACPY( 'All', N, NRHS, B, LDB, X, LDX ) |
CALL ZGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, X, LDX, |
CALL ZGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, X, LDX, |
+ INFO ) |
$ INFO ) |
* |
* |
RETURN |
RETURN |
* |
* |