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DOUBLE PRECISION FUNCTION DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
$ LDAF, IPIV, WORK )
*
* -- LAPACK routine (version 3.2.1) --
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
* -- Jason Riedy of Univ. of California Berkeley. --
* -- April 2009 --
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley and NAG Ltd. --
*
IMPLICIT NONE
* ..
* .. Scalar Arguments ..
CHARACTER*1 UPLO
INTEGER N, INFO, LDA, LDAF
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DLA_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.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* INFO (input) INTEGER
* The value of INFO returned from DSYTRF, .i.e., the pivot in
* column INFO is exactly 0.
*
* NCOLS (input) INTEGER
* The number of columns of the matrix A. NCOLS >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the N-by-N matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
* The block diagonal matrix D and the multipliers used to
* obtain the factor U or L as computed by DSYTRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D
* as determined by DSYTRF.
*
* WORK (input) DOUBLE PRECISION array, dimension (2*N)
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER NCOLS, I, J, K, KP
DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP
LOGICAL UPPER
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
EXTERNAL LSAME, DLASET
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 = 1.0D+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( 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 dsytrs.
* 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 /= 0.0D+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
DLA_SYRPVGRW = RPVGRW
END FUNCTION
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