*> \brief \b ZHEEQUB * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZHEEQUB + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, N * DOUBLE PRECISION AMAX, SCOND * CHARACTER UPLO * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), WORK( * ) * DOUBLE PRECISION S( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZHEEQUB computes row and column scalings intended to equilibrate a *> Hermitian matrix A (with respect to the Euclidean norm) and reduce *> its condition number. The scale factors S are computed by the BIN *> algorithm (see references) so that the scaled matrix B with elements *> B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of *> the smallest possible condition number over all possible diagonal *> scalings. *> \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 order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> The N-by-N Hermitian matrix whose scaling factors are to be *> computed. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] S *> \verbatim *> S is DOUBLE PRECISION array, dimension (N) *> If INFO = 0, S contains the scale factors for A. *> \endverbatim *> *> \param[out] SCOND *> \verbatim *> SCOND is DOUBLE PRECISION *> If INFO = 0, S contains the ratio of the smallest S(i) to *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too *> large nor too small, it is not worth scaling by S. *> \endverbatim *> *> \param[out] AMAX *> \verbatim *> AMAX is DOUBLE PRECISION *> Largest absolute value of any matrix element. If AMAX is *> very close to overflow or very close to underflow, the *> matrix should be scaled. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (2*N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: if INFO = i, the i-th diagonal element is nonpositive. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complex16HEcomputational * *> \par References: * ================ *> *> Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n *> Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n *> DOI 10.1023/B:NUMA.0000016606.32820.69 \n *> Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679 *> * ===================================================================== SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO ) * * -- 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 .. INTEGER INFO, LDA, N DOUBLE PRECISION AMAX, SCOND CHARACTER UPLO * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), WORK( * ) DOUBLE PRECISION S( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 ) INTEGER MAX_ITER PARAMETER ( MAX_ITER = 100 ) * .. * .. Local Scalars .. INTEGER I, J, ITER DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE, \$ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ LOGICAL UP COMPLEX*16 ZDUM * .. * .. External Functions .. DOUBLE PRECISION DLAMCH LOGICAL LSAME EXTERNAL DLAMCH, LSAME * .. * .. External Subroutines .. EXTERNAL ZLASSQ, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function Definitions .. CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -1 ELSE IF ( N .LT. 0 ) THEN INFO = -2 ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN INFO = -4 END IF IF ( INFO .NE. 0 ) THEN CALL XERBLA( 'ZHEEQUB', -INFO ) RETURN END IF UP = LSAME( UPLO, 'U' ) AMAX = ZERO * * Quick return if possible. * IF ( N .EQ. 0 ) THEN SCOND = ONE RETURN END IF DO I = 1, N S( I ) = ZERO END DO AMAX = ZERO IF ( UP ) THEN DO J = 1, N DO I = 1, J-1 S( I ) = MAX( S( I ), CABS1( A( I, J ) ) ) S( J ) = MAX( S( J ), CABS1( A( I, J ) ) ) AMAX = MAX( AMAX, CABS1( A( I, J ) ) ) END DO S( J ) = MAX( S( J ), CABS1( A( J, J ) ) ) AMAX = MAX( AMAX, CABS1( A( J, J ) ) ) END DO ELSE DO J = 1, N S( J ) = MAX( S( J ), CABS1( A( J, J ) ) ) AMAX = MAX( AMAX, CABS1( A( J, J ) ) ) DO I = J+1, N S( I ) = MAX( S( I ), CABS1( A( I, J ) ) ) S( J ) = MAX( S( J ), CABS1( A( I, J ) ) ) AMAX = MAX( AMAX, CABS1( A( I, J ) ) ) END DO END DO END IF DO J = 1, N S( J ) = 1.0D0 / S( J ) END DO TOL = ONE / SQRT( 2.0D0 * N ) DO ITER = 1, MAX_ITER SCALE = 0.0D0 SUMSQ = 0.0D0 * beta = |A|s DO I = 1, N WORK( I ) = ZERO END DO IF ( UP ) THEN DO J = 1, N DO I = 1, J-1 WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J ) WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I ) END DO WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J ) END DO ELSE DO J = 1, N WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J ) DO I = J+1, N WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J ) WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I ) END DO END DO END IF * avg = s^T beta / n AVG = 0.0D0 DO I = 1, N AVG = AVG + DBLE( S( I )*WORK( I ) ) END DO AVG = AVG / N STD = 0.0D0 DO I = N+1, 2*N WORK( I ) = S( I-N ) * WORK( I-N ) - AVG END DO CALL ZLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ ) STD = SCALE * SQRT( SUMSQ / N ) IF ( STD .LT. TOL * AVG ) GOTO 999 DO I = 1, N T = CABS1( A( I, I ) ) SI = S( I ) C2 = ( N-1 ) * T C1 = ( N-2 ) * ( DBLE( WORK( I ) ) - T*SI ) C0 = -(T*SI)*SI + 2 * DBLE( WORK( I ) ) * SI - N*AVG D = C1*C1 - 4*C0*C2 IF ( D .LE. 0 ) THEN INFO = -1 RETURN END IF SI = -2*C0 / ( C1 + SQRT( D ) ) D = SI - S( I ) U = ZERO IF ( UP ) THEN DO J = 1, I T = CABS1( A( J, I ) ) U = U + S( J )*T WORK( J ) = WORK( J ) + D*T END DO DO J = I+1,N T = CABS1( A( I, J ) ) U = U + S( J )*T WORK( J ) = WORK( J ) + D*T END DO ELSE DO J = 1, I T = CABS1( A( I, J ) ) U = U + S( J )*T WORK( J ) = WORK( J ) + D*T END DO DO J = I+1,N T = CABS1( A( J, I ) ) U = U + S( J )*T WORK( J ) = WORK( J ) + D*T END DO END IF AVG = AVG + ( U + DBLE( WORK( I ) ) ) * D / N S( I ) = SI END DO END DO 999 CONTINUE SMLNUM = DLAMCH( 'SAFEMIN' ) BIGNUM = ONE / SMLNUM SMIN = BIGNUM SMAX = ZERO T = ONE / SQRT( AVG ) BASE = DLAMCH( 'B' ) U = ONE / LOG( BASE ) DO I = 1, N S( I ) = BASE ** INT( U * LOG( S( I ) * T ) ) SMIN = MIN( SMIN, S( I ) ) SMAX = MAX( SMAX, S( I ) ) END DO SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) * END