zla_porpvgrw.f

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*> \brief \b ZLA_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/
*
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*
*  Definition:
*  ===========
*
*       DOUBLE PRECISION FUNCTION ZLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF,
*                                               LDAF, WORK )
*
*       .. Scalar Arguments ..
*       CHARACTER*1        UPLO
*       INTEGER            NCOLS, LDA, LDAF
*       ..
*       .. Array Arguments ..
*       COMPLEX*16         A( LDA, * ), AF( LDAF, * )
*       DOUBLE PRECISION   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>
*> ZLA_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 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 triangular factor U or L from the Cholesky factorization
*>     A = U**T*U or A = L*L**T, as computed by ZPOTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>     The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[out] 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.
*
*> \ingroup la_porpvgrw
*
*  =====================================================================
      DOUBLE PRECISION FUNCTION zla_porpvgrw( UPLO, NCOLS, A, LDA, AF,
     $                                        LDAF, WORK )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER*1        uplo
      INTEGER            ncols, lda, ldaf
*     ..
*     .. Array Arguments ..
      COMPLEX*16         a( lda, * ), af( ldaf, * )
      DOUBLE PRECISION   work( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            i, j
      DOUBLE PRECISION   amax, umax, rpvgrw
      LOGICAL            upper
      COMPLEX*16         zdum
*     ..
*     .. External Functions ..
      EXTERNAL           lsame
      LOGICAL            lsame
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, real, dimag
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   cabs1
*     ..
*     .. Statement Function Definitions ..
      cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
*     ..
*     .. Executable Statements ..
      upper = lsame( 'Upper', uplo )
*
*     DPOTRF will have factored only the NCOLSxNCOLS leading submatrix,
*     so we restrict the growth search to that submatrix 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( cabs1( a( i, j ) ), work( ncols+j ) )
            END DO
         END DO
      ELSE
         DO j = 1, ncols
            DO i = j, ncols
               work( ncols+j ) =
     $              max( cabs1( 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( cabs1( af( i, j ) ), work( j ) )
            END DO
         END DO
      ELSE
         DO j = 1, ncols
            DO i = j, ncols
               work( j ) = max( cabs1( 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
 
      zla_porpvgrw = rpvgrw
*
*     End of ZLA_PORPVGRW
*
      END