*> \brief \b DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION TOL
* INTEGER INFO, LDA, N, RANK
* CHARACTER UPLO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
* INTEGER PIV( N )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPSTF2 computes the Cholesky factorization with complete
*> pivoting of a real symmetric positive semidefinite matrix A.
*>
*> The factorization has the form
*> P**T * A * P = U**T * U , if UPLO = 'U',
*> P**T * A * P = L * L**T, 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 2 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 DOUBLE PRECISION 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[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 DOUBLE PRECISION
*> 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[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION 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.
*
*> \date December 2016
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, LDA, N, RANK
CHARACTER UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
INTEGER PIV( N )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AJJ, DSTOP, DTEMP
INTEGER I, ITEMP, J, PVT
LOGICAL UPPER
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
LOGICAL LSAME, DISNAN
EXTERNAL DLAMCH, LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, 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( 'DPSTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Initialize PIV
*
DO 100 I = 1, N
PIV( I ) = I
100 CONTINUE
*
* Compute stopping value
*
PVT = 1
AJJ = A( PVT, PVT )
DO I = 2, N
IF( A( I, I ).GT.AJJ ) THEN
PVT = I
AJJ = A( PVT, PVT )
END IF
END DO
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
RANK = 0
INFO = 1
GO TO 170
END IF
*
* Compute stopping value if not supplied
*
IF( TOL.LT.ZERO ) THEN
DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
ELSE
DSTOP = TOL
END IF
*
* Set first half of WORK to zero, holds dot products
*
DO 110 I = 1, N
WORK( I ) = 0
110 CONTINUE
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization P**T * A * P = U**T * U
*
DO 130 J = 1, N
*
* 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 120 I = J, N
*
IF( J.GT.1 ) THEN
WORK( I ) = WORK( I ) + A( J-1, I )**2
END IF
WORK( N+I ) = A( I, I ) - WORK( I )
*
120 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.DSTOP.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 160
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 DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
IF( PVT.LT.N )
$ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
$ A( PVT, PVT+1 ), LDA )
CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ), 1 )
*
* Swap dot products and PIV
*
DTEMP = WORK( J )
WORK( J ) = WORK( PVT )
WORK( PVT ) = DTEMP
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 DGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
$ A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
END IF
*
130 CONTINUE
*
ELSE
*
* Compute the Cholesky factorization P**T * A * P = L * L**T
*
DO 150 J = 1, N
*
* 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 140 I = J, N
*
IF( J.GT.1 ) THEN
WORK( I ) = WORK( I ) + A( I, J-1 )**2
END IF
WORK( N+I ) = A( I, I ) - WORK( I )
*
140 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.DSTOP.OR.DISNAN( AJJ ) ) THEN
A( J, J ) = AJJ
GO TO 160
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 DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
IF( PVT.LT.N )
$ CALL DSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
$ 1 )
CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), LDA )
*
* Swap dot products and PIV
*
DTEMP = WORK( J )
WORK( J ) = WORK( PVT )
WORK( PVT ) = DTEMP
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 DGEMV( 'No Trans', N-J, J-1, -ONE, A( J+1, 1 ), LDA,
$ A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
END IF
*
150 CONTINUE
*
END IF
*
* Ran to completion, A has full rank
*
RANK = N
*
GO TO 170
160 CONTINUE
*
* Rank is number of steps completed. Set INFO = 1 to signal
* that the factorization cannot be used to solve a system.
*
RANK = J - 1
INFO = 1
*
170 CONTINUE
RETURN
*
* End of DPSTF2
*
END