*> \brief \b ZLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLA_SYRPVGRW + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * DOUBLE PRECISION FUNCTION ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, * LDAF, IPIV, WORK ) * * .. Scalar Arguments .. * CHARACTER*1 UPLO * INTEGER N, INFO, LDA, LDAF * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), AF( LDAF, * ) * DOUBLE PRECISION WORK( * ) * INTEGER IPIV( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> *> ZLA_SYRPVGRW computes the reciprocal pivot growth factor *> norm(A)/norm(U). The "max absolute element" norm is used. If this is *> much less than 1, the stability of the LU factorization of the *> (equilibrated) matrix A could be poor. This also means that the *> solution X, estimated condition numbers, and error bounds could be *> unreliable. *> \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] INFO *> \verbatim *> INFO is INTEGER *> The value of INFO returned from ZSYTRF, .i.e., the pivot in *> column INFO is exactly 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> On entry, the N-by-N matrix A. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] AF *> \verbatim *> AF is COMPLEX*16 array, dimension (LDAF,N) *> The block diagonal matrix D and the multipliers used to *> obtain the factor U or L as computed by ZSYTRF. *> \endverbatim *> *> \param[in] LDAF *> \verbatim *> LDAF is INTEGER *> The leading dimension of the array AF. LDAF >= max(1,N). *> \endverbatim *> *> \param[in] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> Details of the interchanges and the block structure of D *> as determined by ZSYTRF. *> \endverbatim *> *> \param[in] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (2*N) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16SYcomputational * * ===================================================================== DOUBLE PRECISION FUNCTION ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, \$ LDAF, IPIV, WORK ) * * -- 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 .. CHARACTER*1 UPLO INTEGER N, INFO, LDA, LDAF * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), AF( LDAF, * ) DOUBLE PRECISION WORK( * ) INTEGER IPIV( * ) * .. * * ===================================================================== * * .. Local Scalars .. INTEGER NCOLS, I, J, K, KP DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP LOGICAL UPPER COMPLEX*16 ZDUM * .. * .. Intrinsic Functions .. INTRINSIC ABS, REAL, DIMAG, MAX, MIN * .. * .. External Subroutines .. EXTERNAL LSAME LOGICAL LSAME * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function Definitions .. CABS1( ZDUM ) = ABS( DBLE ( ZDUM ) ) + ABS( DIMAG ( ZDUM ) ) * .. * .. Executable Statements .. * UPPER = LSAME( 'Upper', UPLO ) IF ( INFO.EQ.0 ) THEN IF ( UPPER ) THEN NCOLS = 1 ELSE NCOLS = N END IF ELSE NCOLS = INFO END IF RPVGRW = 1.0D+0 DO I = 1, 2*N WORK( I ) = 0.0D+0 END DO * * Find the max magnitude entry of each column of A. Compute the max * for all N columns so we can apply the pivot permutation while * looping below. Assume a full factorization is the common case. * IF ( UPPER ) THEN DO J = 1, N DO I = 1, J WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) ) WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) ) END DO END DO ELSE DO J = 1, N DO I = J, N WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) ) WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) ) END DO END DO END IF * * Now find the max magnitude entry of each column of U or L. Also * permute the magnitudes of A above so they're in the same order as * the factor. * * The iteration orders and permutations were copied from zsytrs. * Calls to SSWAP would be severe overkill. * IF ( UPPER ) THEN K = N DO WHILE ( K .LT. NCOLS .AND. K.GT.0 ) IF ( IPIV( K ).GT.0 ) THEN ! 1x1 pivot KP = IPIV( K ) IF ( KP .NE. K ) THEN TMP = WORK( N+K ) WORK( N+K ) = WORK( N+KP ) WORK( N+KP ) = TMP END IF DO I = 1, K WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) ) END DO K = K - 1 ELSE ! 2x2 pivot KP = -IPIV( K ) TMP = WORK( N+K-1 ) WORK( N+K-1 ) = WORK( N+KP ) WORK( N+KP ) = TMP DO I = 1, K-1 WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) ) WORK( K-1 ) = \$ MAX( CABS1( AF( I, K-1 ) ), WORK( K-1 ) ) END DO WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) ) K = K - 2 END IF END DO K = NCOLS DO WHILE ( K .LE. N ) IF ( IPIV( K ).GT.0 ) THEN KP = IPIV( K ) IF ( KP .NE. K ) THEN TMP = WORK( N+K ) WORK( N+K ) = WORK( N+KP ) WORK( N+KP ) = TMP END IF K = K + 1 ELSE KP = -IPIV( K ) TMP = WORK( N+K ) WORK( N+K ) = WORK( N+KP ) WORK( N+KP ) = TMP K = K + 2 END IF END DO ELSE K = 1 DO WHILE ( K .LE. NCOLS ) IF ( IPIV( K ).GT.0 ) THEN ! 1x1 pivot KP = IPIV( K ) IF ( KP .NE. K ) THEN TMP = WORK( N+K ) WORK( N+K ) = WORK( N+KP ) WORK( N+KP ) = TMP END IF DO I = K, N WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) ) END DO K = K + 1 ELSE ! 2x2 pivot KP = -IPIV( K ) TMP = WORK( N+K+1 ) WORK( N+K+1 ) = WORK( N+KP ) WORK( N+KP ) = TMP DO I = K+1, N WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) ) WORK( K+1 ) = \$ MAX( CABS1( AF( I, K+1 ) ), WORK( K+1 ) ) END DO WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) ) K = K + 2 END IF END DO K = NCOLS DO WHILE ( K .GE. 1 ) IF ( IPIV( K ).GT.0 ) THEN KP = IPIV( K ) IF ( KP .NE. K ) THEN TMP = WORK( N+K ) WORK( N+K ) = WORK( N+KP ) WORK( N+KP ) = TMP END IF K = K - 1 ELSE KP = -IPIV( K ) TMP = WORK( N+K ) WORK( N+K ) = WORK( N+KP ) WORK( N+KP ) = TMP K = K - 2 ENDIF END DO END IF * * Compute the *inverse* of the max element growth factor. Dividing * by zero would imply the largest entry of the factor's column is * zero. Than can happen when either the column of A is zero or * massive pivots made the factor underflow to zero. Neither counts * as growth in itself, so simply ignore terms with zero * denominators. * IF ( UPPER ) THEN DO I = NCOLS, N UMAX = WORK( I ) AMAX = WORK( N+I ) IF ( UMAX /= 0.0D+0 ) THEN RPVGRW = MIN( AMAX / UMAX, RPVGRW ) END IF END DO ELSE DO I = 1, NCOLS UMAX = WORK( I ) AMAX = WORK( N+I ) IF ( UMAX /= 0.0D+0 ) THEN RPVGRW = MIN( AMAX / UMAX, RPVGRW ) END IF END DO END IF ZLA_SYRPVGRW = RPVGRW END