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*> \brief \b DLA_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 DLA_PORPVGRW + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dla_porpvgrw.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dla_porpvgrw.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_porpvgrw.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF,
* LDAF, WORK )
*
* .. Scalar Arguments ..
* CHARACTER*1 UPLO
* INTEGER NCOLS, LDA, LDAF
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*>
*> DLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DPOTRF.
*> \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 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 doublePOcomputational
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLA_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 ..
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION AMAX, UMAX, RPVGRW
LOGICAL UPPER
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
EXTERNAL LSAME
LOGICAL LSAME
* ..
* .. Executable Statements ..
*
UPPER = LSAME( 'Upper', UPLO )
*
* DPOTRF 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 = 1.0D+0
DO I = 1, 2*NCOLS
WORK( I ) = 0.0D+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 /= 0.0D+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 /= 0.0D+0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
END IF
DLA_PORPVGRW = RPVGRW
END
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