SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO ) * * -- LAPACK PROTOTYPE routine (version 3.2.2) -- * Craig Lucas, University of Manchester / NAG Ltd. * October, 2008 * * .. Scalar Arguments .. DOUBLE PRECISION TOL INTEGER INFO, LDA, N, RANK CHARACTER UPLO * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), WORK( 2*N ) INTEGER PIV( N ) * .. * * Purpose * ======= * * DPSTF2 computes the Cholesky factorization with complete * pivoting of a real symmetric positive semidefinite matrix A. * * The factorization has the form * P' * A * P = U' * U , if UPLO = 'U', * P' * A * P = L * L', 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. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * symmetric matrix A is stored. * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) 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. * * PIV (output) INTEGER array, dimension (N) * PIV is such that the nonzero entries are P( PIV(K), K ) = 1. * * RANK (output) INTEGER * The rank of A given by the number of steps the algorithm * completed. * * TOL (input) 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. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * WORK (workspace) DOUBLE PRECISION array, dimension (2*N) * Work space. * * INFO (output) 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 indefinite. See Section 7 of * LAPACK Working Note #161 for further information. * * ===================================================================== * * .. 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.EQ.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' * A * P = U' * 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' * A * P = L * L' * 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