*> \brief \b SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLA_PORPVGRW + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * REAL FUNCTION SLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF, LDAF, WORK ) * * .. Scalar Arguments .. * CHARACTER*1 UPLO * INTEGER NCOLS, LDA, LDAF * .. * .. Array Arguments .. * REAL A( LDA, * ), AF( LDAF, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> *> SLA_PORPVGRW 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] NCOLS *> \verbatim *> NCOLS is INTEGER *> The number of columns of the matrix A. NCOLS >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is REAL 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 REAL array, dimension (LDAF,N) *> The triangular factor U or L from the Cholesky factorization *> A = U**T*U or A = L*L**T, as computed by SPOTRF. *> \endverbatim *> *> \param[in] LDAF *> \verbatim *> LDAF is INTEGER *> The leading dimension of the array AF. LDAF >= max(1,N). *> \endverbatim *> *> \param[in] WORK *> \verbatim *> WORK is REAL 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 realPOcomputational * * ===================================================================== REAL FUNCTION SLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF, LDAF, 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 NCOLS, LDA, LDAF * .. * .. Array Arguments .. REAL A( LDA, * ), AF( LDAF, * ), WORK( * ) * .. * * ===================================================================== * * .. Local Scalars .. INTEGER I, J REAL AMAX, UMAX, RPVGRW LOGICAL UPPER * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. External Functions .. EXTERNAL LSAME LOGICAL LSAME * .. * .. Executable Statements .. * UPPER = LSAME( 'Upper', UPLO ) * * SPOTRF will have factored only the NCOLSxNCOLS leading minor, so * we restrict the growth search to that minor and use only the first * 2*NCOLS workspace entries. * RPVGRW = 1.0 DO I = 1, 2*NCOLS WORK( I ) = 0.0 END DO * * Find the max magnitude entry of each column. * IF ( UPPER ) THEN DO J = 1, NCOLS DO I = 1, J WORK( NCOLS+J ) = \$ MAX( ABS( A( I, J ) ), WORK( NCOLS+J ) ) END DO END DO ELSE DO J = 1, NCOLS DO I = J, NCOLS WORK( NCOLS+J ) = \$ MAX( ABS( A( I, J ) ), WORK( NCOLS+J ) ) END DO END DO END IF * * Now find the max magnitude entry of each column of the factor in * AF. No pivoting, so no permutations. * IF ( LSAME( 'Upper', UPLO ) ) THEN DO J = 1, NCOLS DO I = 1, J WORK( J ) = MAX( ABS( AF( I, J ) ), WORK( J ) ) END DO END DO ELSE DO J = 1, NCOLS DO I = J, NCOLS WORK( J ) = MAX( ABS( AF( I, J ) ), WORK( J ) ) END DO 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 ( LSAME( 'Upper', UPLO ) ) THEN DO I = 1, NCOLS UMAX = WORK( I ) AMAX = WORK( NCOLS+I ) IF ( UMAX /= 0.0 ) THEN RPVGRW = MIN( AMAX / UMAX, RPVGRW ) END IF END DO ELSE DO I = 1, NCOLS UMAX = WORK( I ) AMAX = WORK( NCOLS+I ) IF ( UMAX /= 0.0 ) THEN RPVGRW = MIN( AMAX / UMAX, RPVGRW ) END IF END DO END IF SLA_PORPVGRW = RPVGRW END