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.
461 lines
14 KiB
461 lines
14 KiB
2 years ago
|
*> \brief \b CPSTRF computes the Cholesky factorization with complete pivoting of complex Hermitian positive semidefinite matrix.
|
||
|
|
*
|
||
|
|
* =========== DOCUMENTATION ===========
|
||
|
|
*
|
||
|
|
* Online html documentation available at
|
||
|
|
* http://www.netlib.org/lapack/explore-html/
|
||
|
|
*
|
||
|
|
*> \htmlonly
|
||
|
|
*> Download CPSTRF + dependencies
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpstrf.f">
|
||
|
|
*> [TGZ]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpstrf.f">
|
||
|
|
*> [ZIP]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpstrf.f">
|
||
|
|
*> [TXT]</a>
|
||
|
|
*> \endhtmlonly
|
||
|
|
*
|
||
|
|
* Definition:
|
||
|
|
* ===========
|
||
|
|
*
|
||
|
|
* SUBROUTINE CPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
* REAL TOL
|
||
|
|
* INTEGER INFO, LDA, N, RANK
|
||
|
|
* CHARACTER UPLO
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
* COMPLEX A( LDA, * )
|
||
|
|
* REAL WORK( 2*N )
|
||
|
|
* INTEGER PIV( N )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
*
|
||
|
|
*> \par Purpose:
|
||
|
|
* =============
|
||
|
|
*>
|
||
|
|
*> \verbatim
|
||
|
|
*>
|
||
|
|
*> CPSTRF computes the Cholesky factorization with complete
|
||
|
|
*> pivoting of a complex Hermitian positive semidefinite matrix A.
|
||
|
|
*>
|
||
|
|
*> The factorization has the form
|
||
|
|
*> P**T * A * P = U**H * U , if UPLO = 'U',
|
||
|
|
*> P**T * A * P = L * L**H, if UPLO = 'L',
|
||
|
|
*> where U is an upper triangular matrix and L is lower triangular, and
|
||
|
|
*> P is stored as vector PIV.
|
||
|
|
*>
|
||
|
|
*> This algorithm does not attempt to check that A is positive
|
||
|
|
*> semidefinite. This version of the algorithm calls level 3 BLAS.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Arguments:
|
||
|
|
* ==========
|
||
|
|
*
|
||
|
|
*> \param[in] UPLO
|
||
|
|
*> \verbatim
|
||
|
|
*> UPLO is CHARACTER*1
|
||
|
|
*> Specifies whether the upper or lower triangular part of the
|
||
|
|
*> symmetric matrix A is stored.
|
||
|
|
*> = 'U': Upper triangular
|
||
|
|
*> = 'L': Lower triangular
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] N
|
||
|
|
*> \verbatim
|
||
|
|
*> N is INTEGER
|
||
|
|
*> The order of the matrix A. N >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in,out] A
|
||
|
|
*> \verbatim
|
||
|
|
*> A is COMPLEX array, dimension (LDA,N)
|
||
|
|
*> On entry, the symmetric 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.
|
||
|
|
*>
|
||
|
|
*> On exit, if INFO = 0, the factor U or L from the Cholesky
|
||
|
|
*> factorization as above.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LDA
|
||
|
|
*> \verbatim
|
||
|
|
*> LDA is INTEGER
|
||
|
|
*> The leading dimension of the array A. LDA >= max(1,N).
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] PIV
|
||
|
|
*> \verbatim
|
||
|
|
*> PIV is INTEGER array, dimension (N)
|
||
|
|
*> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] RANK
|
||
|
|
*> \verbatim
|
||
|
|
*> RANK is INTEGER
|
||
|
|
*> The rank of A given by the number of steps the algorithm
|
||
|
|
*> completed.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] TOL
|
||
|
|
*> \verbatim
|
||
|
|
*> TOL is REAL
|
||
|
|
*> User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
|
||
|
|
*> will be used. The algorithm terminates at the (K-1)st step
|
||
|
|
*> if the pivot <= TOL.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] WORK
|
||
|
|
*> \verbatim
|
||
|
|
*> WORK is REAL array, dimension (2*N)
|
||
|
|
*> Work space.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] INFO
|
||
|
|
*> \verbatim
|
||
|
|
*> INFO is INTEGER
|
||
|
|
*> < 0: If INFO = -K, the K-th argument had an illegal value,
|
||
|
|
*> = 0: algorithm completed successfully, and
|
||
|
|
*> > 0: the matrix A is either rank deficient with computed rank
|
||
|
|
*> as returned in RANK, or is not positive semidefinite. See
|
||
|
|
*> Section 7 of LAPACK Working Note #161 for further
|
||
|
|
*> information.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Authors:
|
||
|
|
* ========
|
||
|
|
*
|
||
|
|
*> \author Univ. of Tennessee
|
||
|
|
*> \author Univ. of California Berkeley
|
||
|
|
*> \author Univ. of Colorado Denver
|
||
|
|
*> \author NAG Ltd.
|
||
|
|
*
|
||
|
|
*> \ingroup complexOTHERcomputational
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
SUBROUTINE CPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
|
||
|
|
*
|
||
|
|
* -- LAPACK computational routine --
|
||
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
REAL TOL
|
||
|
|
INTEGER INFO, LDA, N, RANK
|
||
|
|
CHARACTER UPLO
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
COMPLEX A( LDA, * )
|
||
|
|
REAL WORK( 2*N )
|
||
|
|
INTEGER PIV( N )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
*
|
||
|
|
* .. Parameters ..
|
||
|
|
REAL ONE, ZERO
|
||
|
|
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
|
||
|
|
COMPLEX CONE
|
||
|
|
PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) )
|
||
|
|
* ..
|
||
|
|
* .. Local Scalars ..
|
||
|
|
COMPLEX CTEMP
|
||
|
|
REAL AJJ, SSTOP, STEMP
|
||
|
|
INTEGER I, ITEMP, J, JB, K, NB, PVT
|
||
|
|
LOGICAL UPPER
|
||
|
|
* ..
|
||
|
|
* .. External Functions ..
|
||
|
|
REAL SLAMCH
|
||
|
|
INTEGER ILAENV
|
||
|
|
LOGICAL LSAME, SISNAN
|
||
|
|
EXTERNAL SLAMCH, ILAENV, LSAME, SISNAN
|
||
|
|
* ..
|
||
|
|
* .. External Subroutines ..
|
||
|
|
EXTERNAL CGEMV, CHERK, CLACGV, CPSTF2, CSSCAL, CSWAP,
|
||
|
|
$ XERBLA
|
||
|
|
* ..
|
||
|
|
* .. Intrinsic Functions ..
|
||
|
|
INTRINSIC CONJG, MAX, MIN, REAL, SQRT, MAXLOC
|
||
|
|
* ..
|
||
|
|
* .. Executable Statements ..
|
||
|
|
*
|
||
|
|
* Test the input parameters.
|
||
|
|
*
|
||
|
|
INFO = 0
|
||
|
|
UPPER = LSAME( UPLO, 'U' )
|
||
|
|
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
|
||
|
|
INFO = -1
|
||
|
|
ELSE IF( N.LT.0 ) THEN
|
||
|
|
INFO = -2
|
||
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||
|
|
INFO = -4
|
||
|
|
END IF
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
CALL XERBLA( 'CPSTRF', -INFO )
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Quick return if possible
|
||
|
|
*
|
||
|
|
IF( N.EQ.0 )
|
||
|
|
$ RETURN
|
||
|
|
*
|
||
|
|
* Get block size
|
||
|
|
*
|
||
|
|
NB = ILAENV( 1, 'CPOTRF', UPLO, N, -1, -1, -1 )
|
||
|
|
IF( NB.LE.1 .OR. NB.GE.N ) THEN
|
||
|
|
*
|
||
|
|
* Use unblocked code
|
||
|
|
*
|
||
|
|
CALL CPSTF2( UPLO, N, A( 1, 1 ), LDA, PIV, RANK, TOL, WORK,
|
||
|
|
$ INFO )
|
||
|
|
GO TO 230
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* Initialize PIV
|
||
|
|
*
|
||
|
|
DO 100 I = 1, N
|
||
|
|
PIV( I ) = I
|
||
|
|
100 CONTINUE
|
||
|
|
*
|
||
|
|
* Compute stopping value
|
||
|
|
*
|
||
|
|
DO 110 I = 1, N
|
||
|
|
WORK( I ) = REAL( A( I, I ) )
|
||
|
|
110 CONTINUE
|
||
|
|
PVT = MAXLOC( WORK( 1:N ), 1 )
|
||
|
|
AJJ = REAL( A( PVT, PVT ) )
|
||
|
|
IF( AJJ.LE.ZERO.OR.SISNAN( AJJ ) ) THEN
|
||
|
|
RANK = 0
|
||
|
|
INFO = 1
|
||
|
|
GO TO 230
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Compute stopping value if not supplied
|
||
|
|
*
|
||
|
|
IF( TOL.LT.ZERO ) THEN
|
||
|
|
SSTOP = N * SLAMCH( 'Epsilon' ) * AJJ
|
||
|
|
ELSE
|
||
|
|
SSTOP = TOL
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
*
|
||
|
|
IF( UPPER ) THEN
|
||
|
|
*
|
||
|
|
* Compute the Cholesky factorization P**T * A * P = U**H * U
|
||
|
|
*
|
||
|
|
DO 160 K = 1, N, NB
|
||
|
|
*
|
||
|
|
* Account for last block not being NB wide
|
||
|
|
*
|
||
|
|
JB = MIN( NB, N-K+1 )
|
||
|
|
*
|
||
|
|
* Set relevant part of first half of WORK to zero,
|
||
|
|
* holds dot products
|
||
|
|
*
|
||
|
|
DO 120 I = K, N
|
||
|
|
WORK( I ) = 0
|
||
|
|
120 CONTINUE
|
||
|
|
*
|
||
|
|
DO 150 J = K, K + JB - 1
|
||
|
|
*
|
||
|
|
* Find pivot, test for exit, else swap rows and columns
|
||
|
|
* Update dot products, compute possible pivots which are
|
||
|
|
* stored in the second half of WORK
|
||
|
|
*
|
||
|
|
DO 130 I = J, N
|
||
|
|
*
|
||
|
|
IF( J.GT.K ) THEN
|
||
|
|
WORK( I ) = WORK( I ) +
|
||
|
|
$ REAL( CONJG( A( J-1, I ) )*
|
||
|
|
$ A( J-1, I ) )
|
||
|
|
END IF
|
||
|
|
WORK( N+I ) = REAL( A( I, I ) ) - WORK( I )
|
||
|
|
*
|
||
|
|
130 CONTINUE
|
||
|
|
*
|
||
|
|
IF( J.GT.1 ) THEN
|
||
|
|
ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
|
||
|
|
PVT = ITEMP + J - 1
|
||
|
|
AJJ = WORK( N+PVT )
|
||
|
|
IF( AJJ.LE.SSTOP.OR.SISNAN( AJJ ) ) THEN
|
||
|
|
A( J, J ) = AJJ
|
||
|
|
GO TO 220
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( J.NE.PVT ) THEN
|
||
|
|
*
|
||
|
|
* Pivot OK, so can now swap pivot rows and columns
|
||
|
|
*
|
||
|
|
A( PVT, PVT ) = A( J, J )
|
||
|
|
CALL CSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
|
||
|
|
IF( PVT.LT.N )
|
||
|
|
$ CALL CSWAP( N-PVT, A( J, PVT+1 ), LDA,
|
||
|
|
$ A( PVT, PVT+1 ), LDA )
|
||
|
|
DO 140 I = J + 1, PVT - 1
|
||
|
|
CTEMP = CONJG( A( J, I ) )
|
||
|
|
A( J, I ) = CONJG( A( I, PVT ) )
|
||
|
|
A( I, PVT ) = CTEMP
|
||
|
|
140 CONTINUE
|
||
|
|
A( J, PVT ) = CONJG( A( J, PVT ) )
|
||
|
|
*
|
||
|
|
* Swap dot products and PIV
|
||
|
|
*
|
||
|
|
STEMP = WORK( J )
|
||
|
|
WORK( J ) = WORK( PVT )
|
||
|
|
WORK( PVT ) = STEMP
|
||
|
|
ITEMP = PIV( PVT )
|
||
|
|
PIV( PVT ) = PIV( J )
|
||
|
|
PIV( J ) = ITEMP
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
AJJ = SQRT( AJJ )
|
||
|
|
A( J, J ) = AJJ
|
||
|
|
*
|
||
|
|
* Compute elements J+1:N of row J.
|
||
|
|
*
|
||
|
|
IF( J.LT.N ) THEN
|
||
|
|
CALL CLACGV( J-1, A( 1, J ), 1 )
|
||
|
|
CALL CGEMV( 'Trans', J-K, N-J, -CONE, A( K, J+1 ),
|
||
|
|
$ LDA, A( K, J ), 1, CONE, A( J, J+1 ),
|
||
|
|
$ LDA )
|
||
|
|
CALL CLACGV( J-1, A( 1, J ), 1 )
|
||
|
|
CALL CSSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
150 CONTINUE
|
||
|
|
*
|
||
|
|
* Update trailing matrix, J already incremented
|
||
|
|
*
|
||
|
|
IF( K+JB.LE.N ) THEN
|
||
|
|
CALL CHERK( 'Upper', 'Conj Trans', N-J+1, JB, -ONE,
|
||
|
|
$ A( K, J ), LDA, ONE, A( J, J ), LDA )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
160 CONTINUE
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* Compute the Cholesky factorization P**T * A * P = L * L**H
|
||
|
|
*
|
||
|
|
DO 210 K = 1, N, NB
|
||
|
|
*
|
||
|
|
* Account for last block not being NB wide
|
||
|
|
*
|
||
|
|
JB = MIN( NB, N-K+1 )
|
||
|
|
*
|
||
|
|
* Set relevant part of first half of WORK to zero,
|
||
|
|
* holds dot products
|
||
|
|
*
|
||
|
|
DO 170 I = K, N
|
||
|
|
WORK( I ) = 0
|
||
|
|
170 CONTINUE
|
||
|
|
*
|
||
|
|
DO 200 J = K, K + JB - 1
|
||
|
|
*
|
||
|
|
* Find pivot, test for exit, else swap rows and columns
|
||
|
|
* Update dot products, compute possible pivots which are
|
||
|
|
* stored in the second half of WORK
|
||
|
|
*
|
||
|
|
DO 180 I = J, N
|
||
|
|
*
|
||
|
|
IF( J.GT.K ) THEN
|
||
|
|
WORK( I ) = WORK( I ) +
|
||
|
|
$ REAL( CONJG( A( I, J-1 ) )*
|
||
|
|
$ A( I, J-1 ) )
|
||
|
|
END IF
|
||
|
|
WORK( N+I ) = REAL( A( I, I ) ) - WORK( I )
|
||
|
|
*
|
||
|
|
180 CONTINUE
|
||
|
|
*
|
||
|
|
IF( J.GT.1 ) THEN
|
||
|
|
ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
|
||
|
|
PVT = ITEMP + J - 1
|
||
|
|
AJJ = WORK( N+PVT )
|
||
|
|
IF( AJJ.LE.SSTOP.OR.SISNAN( AJJ ) ) THEN
|
||
|
|
A( J, J ) = AJJ
|
||
|
|
GO TO 220
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( J.NE.PVT ) THEN
|
||
|
|
*
|
||
|
|
* Pivot OK, so can now swap pivot rows and columns
|
||
|
|
*
|
||
|
|
A( PVT, PVT ) = A( J, J )
|
||
|
|
CALL CSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
|
||
|
|
IF( PVT.LT.N )
|
||
|
|
$ CALL CSWAP( N-PVT, A( PVT+1, J ), 1,
|
||
|
|
$ A( PVT+1, PVT ), 1 )
|
||
|
|
DO 190 I = J + 1, PVT - 1
|
||
|
|
CTEMP = CONJG( A( I, J ) )
|
||
|
|
A( I, J ) = CONJG( A( PVT, I ) )
|
||
|
|
A( PVT, I ) = CTEMP
|
||
|
|
190 CONTINUE
|
||
|
|
A( PVT, J ) = CONJG( A( PVT, J ) )
|
||
|
|
*
|
||
|
|
* Swap dot products and PIV
|
||
|
|
*
|
||
|
|
STEMP = WORK( J )
|
||
|
|
WORK( J ) = WORK( PVT )
|
||
|
|
WORK( PVT ) = STEMP
|
||
|
|
ITEMP = PIV( PVT )
|
||
|
|
PIV( PVT ) = PIV( J )
|
||
|
|
PIV( J ) = ITEMP
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
AJJ = SQRT( AJJ )
|
||
|
|
A( J, J ) = AJJ
|
||
|
|
*
|
||
|
|
* Compute elements J+1:N of column J.
|
||
|
|
*
|
||
|
|
IF( J.LT.N ) THEN
|
||
|
|
CALL CLACGV( J-1, A( J, 1 ), LDA )
|
||
|
|
CALL CGEMV( 'No Trans', N-J, J-K, -CONE,
|
||
|
|
$ A( J+1, K ), LDA, A( J, K ), LDA, CONE,
|
||
|
|
$ A( J+1, J ), 1 )
|
||
|
|
CALL CLACGV( J-1, A( J, 1 ), LDA )
|
||
|
|
CALL CSSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
200 CONTINUE
|
||
|
|
*
|
||
|
|
* Update trailing matrix, J already incremented
|
||
|
|
*
|
||
|
|
IF( K+JB.LE.N ) THEN
|
||
|
|
CALL CHERK( 'Lower', 'No Trans', N-J+1, JB, -ONE,
|
||
|
|
$ A( J, K ), LDA, ONE, A( J, J ), LDA )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
210 CONTINUE
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Ran to completion, A has full rank
|
||
|
|
*
|
||
|
|
RANK = N
|
||
|
|
*
|
||
|
|
GO TO 230
|
||
|
|
220 CONTINUE
|
||
|
|
*
|
||
|
|
* Rank is the number of steps completed. Set INFO = 1 to signal
|
||
|
|
* that the factorization cannot be used to solve a system.
|
||
|
|
*
|
||
|
|
RANK = J - 1
|
||
|
|
INFO = 1
|
||
|
|
*
|
||
|
|
230 CONTINUE
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of CPSTRF
|
||
|
|
*
|
||
|
|
END
|