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*> \brief \b ZHEEQUB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> Download ZHEEQUB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheequb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheequb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheequb.f">
*> [TXT]</a>
*
* 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( * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> ZSYEQUB computes row and column scalings intended to equilibrate a
*> symmetric matrix A and reduce its condition number
*> (with respect to the two-norm). S contains the scale factors,
*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
*> choice of S puts the condition number of B within a factor N of the
*> smallest possible condition number over all possible diagonal
*> scalings.
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \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 symmetric matrix whose scaling
*> factors are to be computed. Only the diagonal elements of A
*> are referenced.
*> \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
*> Absolute value of largest matrix element. If AMAX is very
*> close to overflow or very close to underflow, the matrix
*> should be scaled.
*> \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
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex16HEcomputational
*
* =====================================================================
SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
*
* -- LAPACK computational routine (version 3.2.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. 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 = 1.0D+0, ZERO = 0.0D+0 )
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
* ..
* .. 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 ) )
*
* Test 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 ) = 1.0D+0 / S( J )
END DO
TOL = ONE / SQRT( 2.0D0 * N )
DO ITER = 1, MAX_ITER
SCALE = 0.0D+0
SUMSQ = 0.0D+0
* beta = |A|s
DO I = 1, N
WORK( I ) = ZERO
END DO
IF ( UP ) THEN
DO J = 1, N
DO I = 1, J-1
T = CABS1( A( I, J ) )
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
T = CABS1( A( I, J ) )
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 = 0.0D+0
DO I = 1, N
AVG = AVG + S( I )*WORK( I )
END DO
AVG = AVG / N
STD = 0.0D+0
DO I = 2*N+1, 3*N
WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
END DO
CALL ZLASSQ( N, WORK( 2*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 ) * ( WORK( I ) - T*SI )
C0 = -(T*SI)*SI + 2*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 + 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
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