*> \brief \b SLA_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/
*
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*
* Definition:
* ===========
*
* REAL FUNCTION SLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV,
* WORK )
*
* .. Scalar Arguments ..
* CHARACTER*1 UPLO
* INTEGER N, INFO, LDA, LDAF
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* REAL A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*>
*> SLA_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 SSYTRF, .i.e., the pivot in
*> column INFO is exactly 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 block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by SSYTRF.
*> \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 SSYTRF.
*> \endverbatim
*>
*> \param[out] 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 realSYcomputational
*
* =====================================================================
REAL FUNCTION SLA_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 ..
INTEGER IPIV( * )
REAL A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER NCOLS, I, J, K, KP
REAL AMAX, UMAX, RPVGRW, TMP
LOGICAL UPPER
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
EXTERNAL LSAME
LOGICAL LSAME
* ..
* .. 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 = 1.0
DO I = 1, 2*N
WORK( I ) = 0.0
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( ABS( A( I, J ) ), WORK( N+I ) )
WORK( N+J ) = MAX( ABS( A( I, J ) ), WORK( N+J ) )
END DO
END DO
ELSE
DO J = 1, N
DO I = J, N
WORK( N+I ) = MAX( ABS( A( I, J ) ), WORK( N+I ) )
WORK( N+J ) = MAX( ABS( 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 ssytrs.
* 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( ABS( 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( ABS( AF( I, K ) ), WORK( K ) )
WORK( K-1 ) = MAX( ABS( AF( I, K-1 ) ), WORK( K-1 ) )
END DO
WORK( K ) = MAX( ABS( 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( ABS( 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( ABS( AF( I, K ) ), WORK( K ) )
WORK( K+1 ) = MAX( ABS( AF(I, K+1 ) ), WORK( K+1 ) )
END DO
WORK( K ) = MAX( ABS( 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 /= 0.0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
ELSE
DO I = 1, NCOLS
UMAX = WORK( I )
AMAX = WORK( N+I )
IF ( UMAX /= 0.0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
END IF
SLA_SYRPVGRW = RPVGRW
END