DOUBLE PRECISION FUNCTION ZLA_HERPVGRW( 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( * ) COMPLEX*16 A( LDA, * ), AF( LDAF, * ) DOUBLE PRECISION WORK( * ) * .. * * Purpose * ======= * * ZLA_HERPVGRW 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 ZHETRF, .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) COMPLEX*16 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) COMPLEX*16 array, dimension (LDAF,N) * The block diagonal matrix D and the multipliers used to * obtain the factor U or L as computed by ZHETRF. * * 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 ZHETRF. * * WORK (input) COMPLEX*16 array, dimension (2*N) * * ===================================================================== * * .. Local Scalars .. INTEGER NCOLS, I, J, K, KP DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP LOGICAL UPPER, LSAME COMPLEX*16 ZDUM * .. * .. External Functions .. EXTERNAL LSAME, ZLASET * .. * .. Intrinsic Functions .. INTRINSIC ABS, REAL, DIMAG, MAX, MIN * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function Definitions .. CABS1( ZDUM ) = ABS( DBLE ( ZDUM ) ) + ABS( DIMAG ( ZDUM ) ) * .. * .. 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( CABS1( A( I,J ) ), WORK( N+I ) ) WORK( N+J ) = MAX( CABS1( A( I,J ) ), WORK( N+J ) ) END DO END DO ELSE DO J = 1, N DO I = J, N WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) ) WORK( N+J ) = MAX( CABS1( 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 zsytrs. * 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( CABS1( 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( CABS1( AF( I, K ) ), WORK( K ) ) WORK( K-1 ) = \$ MAX( CABS1( AF( I, K-1 ) ), WORK( K-1 ) ) END DO WORK( K ) = MAX( CABS1( 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( CABS1( 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( CABS1( AF( I, K ) ), WORK( K ) ) WORK( K+1 ) = \$ MAX( CABS1( AF( I, K+1 ) ) , WORK( K+1 ) ) END DO WORK(K) = MAX( CABS1( 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 ZLA_HERPVGRW = RPVGRW END FUNCTION