You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
454 lines
14 KiB
454 lines
14 KiB
*> \brief <b> ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download ZCPOSV + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zcposv.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zcposv.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zcposv.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
|
|
* SWORK, RWORK, ITER, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER UPLO
|
|
* INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* DOUBLE PRECISION RWORK( * )
|
|
* COMPLEX SWORK( * )
|
|
* COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ),
|
|
* $ X( LDX, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> ZCPOSV computes the solution to a complex system of linear equations
|
|
*> A * X = B,
|
|
*> where A is an N-by-N Hermitian positive definite matrix and X and B
|
|
*> are N-by-NRHS matrices.
|
|
*>
|
|
*> ZCPOSV 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] UPLO
|
|
*> \verbatim
|
|
*> UPLO is CHARACTER*1
|
|
*> = 'U': Upper triangle of A is stored;
|
|
*> = 'L': Lower triangle of A is stored.
|
|
*> \endverbatim
|
|
*>
|
|
*> \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 Hermitian matrix A. If UPLO = 'U', the leading
|
|
*> N-by-N upper triangular part of A contains the upper
|
|
*> triangular part of the matrix A, and the strictly lower
|
|
*> triangular part of A is not referenced. If UPLO = 'L', the
|
|
*> leading N-by-N lower triangular part of A contains the lower
|
|
*> triangular part of the matrix A, and the strictly upper
|
|
*> triangular part of A is not referenced.
|
|
*>
|
|
*> Note that the imaginary parts of the diagonal
|
|
*> elements need not be set and are assumed to be zero.
|
|
*>
|
|
*> On exit, if iterative refinement has been successfully used
|
|
*> (INFO = 0 and ITER >= 0, see description below), then A is
|
|
*> unchanged, if double precision factorization has been used
|
|
*> (INFO = 0 and ITER < 0, see description below), then the
|
|
*> array A contains the factor U or L from the Cholesky
|
|
*> factorization A = U**H*U or A = L*L**H.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDA
|
|
*> \verbatim
|
|
*> LDA is INTEGER
|
|
*> The leading dimension of the array A. LDA >= max(1,N).
|
|
*> \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 CPOTRF
|
|
*> -31: stop the iterative refinement after the 30th
|
|
*> iterations
|
|
*> > 0: iterative refinement has been successfully 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, the leading principal minor of order i
|
|
*> of (COMPLEX*16) A is not positive, so the factorization
|
|
*> could not be completed, and the solution has not been
|
|
*> computed.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \ingroup complex16POsolve
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
|
|
$ SWORK, RWORK, ITER, INFO )
|
|
*
|
|
* -- LAPACK driver routine --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER UPLO
|
|
INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
DOUBLE PRECISION RWORK( * )
|
|
COMPLEX SWORK( * )
|
|
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ),
|
|
$ X( LDX, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
LOGICAL DOITREF
|
|
PARAMETER ( DOITREF = .TRUE. )
|
|
*
|
|
INTEGER ITERMAX
|
|
PARAMETER ( ITERMAX = 30 )
|
|
*
|
|
DOUBLE PRECISION BWDMAX
|
|
PARAMETER ( BWDMAX = 1.0E+00 )
|
|
*
|
|
COMPLEX*16 NEGONE, ONE
|
|
PARAMETER ( NEGONE = ( -1.0D+00, 0.0D+00 ),
|
|
$ ONE = ( 1.0D+00, 0.0D+00 ) )
|
|
*
|
|
* .. Local Scalars ..
|
|
INTEGER I, IITER, PTSA, PTSX
|
|
DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM
|
|
COMPLEX*16 ZDUM
|
|
*
|
|
* .. External Subroutines ..
|
|
EXTERNAL ZAXPY, ZHEMM, ZLACPY, ZLAT2C, ZLAG2C, CLAG2Z,
|
|
$ CPOTRF, CPOTRS, XERBLA, ZPOTRF, ZPOTRS
|
|
* ..
|
|
* .. External Functions ..
|
|
INTEGER IZAMAX
|
|
DOUBLE PRECISION DLAMCH, ZLANHE
|
|
LOGICAL LSAME
|
|
EXTERNAL IZAMAX, DLAMCH, ZLANHE, LSAME
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC ABS, DBLE, MAX, SQRT
|
|
* .. Statement Functions ..
|
|
DOUBLE PRECISION CABS1
|
|
* ..
|
|
* .. Statement Function definitions ..
|
|
CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
INFO = 0
|
|
ITER = 0
|
|
*
|
|
* Test the input parameters.
|
|
*
|
|
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
|
|
INFO = -1
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -7
|
|
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
|
|
INFO = -9
|
|
END IF
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZCPOSV', -INFO )
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if (N.EQ.0).
|
|
*
|
|
IF( N.EQ.0 )
|
|
$ RETURN
|
|
*
|
|
* Skip single precision iterative refinement if a priori slower
|
|
* than double precision factorization.
|
|
*
|
|
IF( .NOT.DOITREF ) THEN
|
|
ITER = -1
|
|
GO TO 40
|
|
END IF
|
|
*
|
|
* Compute some constants.
|
|
*
|
|
ANRM = ZLANHE( 'I', UPLO, N, A, LDA, RWORK )
|
|
EPS = DLAMCH( 'Epsilon' )
|
|
CTE = ANRM*EPS*SQRT( DBLE( N ) )*BWDMAX
|
|
*
|
|
* Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
|
|
*
|
|
PTSA = 1
|
|
PTSX = PTSA + N*N
|
|
*
|
|
* Convert B from double precision to single precision and store the
|
|
* result in SX.
|
|
*
|
|
CALL ZLAG2C( N, NRHS, B, LDB, SWORK( PTSX ), N, INFO )
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
ITER = -2
|
|
GO TO 40
|
|
END IF
|
|
*
|
|
* Convert A from double precision to single precision and store the
|
|
* result in SA.
|
|
*
|
|
CALL ZLAT2C( UPLO, N, A, LDA, SWORK( PTSA ), N, INFO )
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
ITER = -2
|
|
GO TO 40
|
|
END IF
|
|
*
|
|
* Compute the Cholesky factorization of SA.
|
|
*
|
|
CALL CPOTRF( UPLO, N, SWORK( PTSA ), N, INFO )
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
ITER = -3
|
|
GO TO 40
|
|
END IF
|
|
*
|
|
* Solve the system SA*SX = SB.
|
|
*
|
|
CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
|
|
$ INFO )
|
|
*
|
|
* Convert SX back to COMPLEX*16
|
|
*
|
|
CALL CLAG2Z( N, NRHS, SWORK( PTSX ), N, X, LDX, INFO )
|
|
*
|
|
* Compute R = B - AX (R is WORK).
|
|
*
|
|
CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )
|
|
*
|
|
CALL ZHEMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
|
|
$ WORK, N )
|
|
*
|
|
* Check whether the NRHS normwise backward errors satisfy the
|
|
* stopping criterion. If yes, set ITER=0 and return.
|
|
*
|
|
DO I = 1, NRHS
|
|
XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )
|
|
RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )
|
|
IF( RNRM.GT.XNRM*CTE )
|
|
$ GO TO 10
|
|
END DO
|
|
*
|
|
* If we are here, the NRHS normwise backward errors satisfy the
|
|
* stopping criterion. We are good to exit.
|
|
*
|
|
ITER = 0
|
|
RETURN
|
|
*
|
|
10 CONTINUE
|
|
*
|
|
DO 30 IITER = 1, ITERMAX
|
|
*
|
|
* Convert R (in WORK) from double precision to single precision
|
|
* and store the result in SX.
|
|
*
|
|
CALL ZLAG2C( N, NRHS, WORK, N, SWORK( PTSX ), N, INFO )
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
ITER = -2
|
|
GO TO 40
|
|
END IF
|
|
*
|
|
* Solve the system SA*SX = SR.
|
|
*
|
|
CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
|
|
$ INFO )
|
|
*
|
|
* Convert SX back to double precision and update the current
|
|
* iterate.
|
|
*
|
|
CALL CLAG2Z( N, NRHS, SWORK( PTSX ), N, WORK, N, INFO )
|
|
*
|
|
DO I = 1, NRHS
|
|
CALL ZAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 )
|
|
END DO
|
|
*
|
|
* Compute R = B - AX (R is WORK).
|
|
*
|
|
CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )
|
|
*
|
|
CALL ZHEMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
|
|
$ WORK, N )
|
|
*
|
|
* Check whether the NRHS normwise backward errors satisfy the
|
|
* stopping criterion. If yes, set ITER=IITER>0 and return.
|
|
*
|
|
DO I = 1, NRHS
|
|
XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )
|
|
RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )
|
|
IF( RNRM.GT.XNRM*CTE )
|
|
$ GO TO 20
|
|
END DO
|
|
*
|
|
* If we are here, the NRHS normwise backward errors satisfy the
|
|
* stopping criterion, we are good to exit.
|
|
*
|
|
ITER = IITER
|
|
*
|
|
RETURN
|
|
*
|
|
20 CONTINUE
|
|
*
|
|
30 CONTINUE
|
|
*
|
|
* If we are at this place of the code, this is because we have
|
|
* performed ITER=ITERMAX iterations and never satisfied the
|
|
* stopping criterion, set up the ITER flag accordingly and follow
|
|
* up on double precision routine.
|
|
*
|
|
ITER = -ITERMAX - 1
|
|
*
|
|
40 CONTINUE
|
|
*
|
|
* Single-precision iterative refinement failed to converge to a
|
|
* satisfactory solution, so we resort to double precision.
|
|
*
|
|
CALL ZPOTRF( UPLO, N, A, LDA, INFO )
|
|
*
|
|
IF( INFO.NE.0 )
|
|
$ RETURN
|
|
*
|
|
CALL ZLACPY( 'All', N, NRHS, B, LDB, X, LDX )
|
|
CALL ZPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO )
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZCPOSV
|
|
*
|
|
END
|
|
|