*> \brief \b SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SPSTRF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO ) * * .. Scalar Arguments .. * REAL TOL * INTEGER INFO, LDA, N, RANK * CHARACTER UPLO * .. * .. Array Arguments .. * REAL A( LDA, * ), WORK( 2*N ) * INTEGER PIV( N ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SPSTRF 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 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 REAL 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. * *> \date December 2016 * *> \ingroup realOTHERcomputational * * ===================================================================== SUBROUTINE SPSTRF( 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 .. REAL TOL INTEGER INFO, LDA, N, RANK CHARACTER UPLO * .. * .. Array Arguments .. REAL A( LDA, * ), WORK( 2*N ) INTEGER PIV( N ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. 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 SGEMV, SPSTF2, SSCAL, SSWAP, SSYRK, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, 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( 'SPSTRF', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Get block size * NB = ILAENV( 1, 'SPOTRF', UPLO, N, -1, -1, -1 ) IF( NB.LE.1 .OR. NB.GE.N ) THEN * * Use unblocked code * CALL SPSTF2( UPLO, N, A( 1, 1 ), LDA, PIV, RANK, TOL, WORK, $ INFO ) GO TO 200 * ELSE * * 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.SISNAN( AJJ ) ) THEN RANK = 0 INFO = 1 GO TO 200 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**T * U * DO 140 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 110 I = K, N WORK( I ) = 0 110 CONTINUE * DO 130 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 120 I = J, N * IF( J.GT.K ) 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.SSTOP.OR.SISNAN( AJJ ) ) THEN A( J, J ) = AJJ GO TO 190 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 SSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 ) IF( PVT.LT.N ) $ CALL SSWAP( N-PVT, A( J, PVT+1 ), LDA, $ A( PVT, PVT+1 ), LDA ) CALL SSWAP( PVT-J-1, A( J, J+1 ), LDA, $ A( J+1, PVT ), 1 ) * * 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 SGEMV( 'Trans', J-K, N-J, -ONE, A( K, J+1 ), $ LDA, A( K, J ), 1, ONE, A( J, J+1 ), $ LDA ) CALL SSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA ) END IF * 130 CONTINUE * * Update trailing matrix, J already incremented * IF( K+JB.LE.N ) THEN CALL SSYRK( 'Upper', 'Trans', N-J+1, JB, -ONE, $ A( K, J ), LDA, ONE, A( J, J ), LDA ) END IF * 140 CONTINUE * ELSE * * Compute the Cholesky factorization P**T * A * P = L * L**T * DO 180 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 150 I = K, N WORK( I ) = 0 150 CONTINUE * DO 170 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 160 I = J, N * IF( J.GT.K ) THEN WORK( I ) = WORK( I ) + A( I, J-1 )**2 END IF WORK( N+I ) = A( I, I ) - WORK( I ) * 160 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 190 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 SSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA ) IF( PVT.LT.N ) $ CALL SSWAP( N-PVT, A( PVT+1, J ), 1, $ A( PVT+1, PVT ), 1 ) CALL SSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), $ LDA ) * * 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 SGEMV( 'No Trans', N-J, J-K, -ONE, $ A( J+1, K ), LDA, A( J, K ), LDA, ONE, $ A( J+1, J ), 1 ) CALL SSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 ) END IF * 170 CONTINUE * * Update trailing matrix, J already incremented * IF( K+JB.LE.N ) THEN CALL SSYRK( 'Lower', 'No Trans', N-J+1, JB, -ONE, $ A( J, K ), LDA, ONE, A( J, J ), LDA ) END IF * 180 CONTINUE * END IF END IF * * Ran to completion, A has full rank * RANK = N * GO TO 200 190 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 * 200 CONTINUE RETURN * * End of SPSTRF * END