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SUBROUTINE CSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
* -- Jason Riedy of Univ. of California Berkeley. --
* -- November 2008 --
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley and NAG Ltd. --
*
IMPLICIT NONE
* ..
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
REAL AMAX, SCOND
CHARACTER UPLO
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), WORK( * )
REAL S( * )
* ..
*
* Purpose
* =======
*
* CSYEQUB 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.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the details of the factorization are stored
* as an upper or lower triangular matrix.
* = 'U': Upper triangular, form is A = U*D*U**T;
* = 'L': Lower triangular, form is A = L*D*L**T.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input) COMPLEX 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* S (output) REAL array, dimension (N)
* If INFO = 0, S contains the scale factors for A.
*
* SCOND (output) REAL
* 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.
*
* AMAX (output) REAL
* Absolute value of largest matrix element. If AMAX is very
* close to overflow or very close to underflow, the matrix
* should be scaled.
*
* WORK (workspace) COMPLEX array, dimension (3*N)
*
* INFO (output) 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.
*
* Further Details
* ======= =======
*
* Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization",
* Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
* DOI 10.1023/B:NUMA.0000016606.32820.69
* Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E0, ZERO = 0.0E0 )
INTEGER MAX_ITER
PARAMETER ( MAX_ITER = 100 )
* ..
* .. Local Scalars ..
INTEGER I, J, ITER
REAL AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
$ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
LOGICAL UP
COMPLEX ZDUM
* ..
* .. External Functions ..
REAL SLAMCH
LOGICAL LSAME
EXTERNAL LSAME, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CLASSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, INT, LOG, MAX, MIN, REAL, SQRT
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* Statement Function Definitions
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( 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( 'CSYEQUB', -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 ) = 1.0 / S( J )
END DO
TOL = ONE / SQRT( 2.0E0 * N )
DO ITER = 1, MAX_ITER
SCALE = 0.0
SUMSQ = 0.0
* 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 = 0.0
DO I = 1, N
AVG = AVG + S( I )*WORK( I )
END DO
AVG = AVG / N
STD = 0.0
DO I = N+1, 2*N
WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
END DO
CALL CLASSQ( 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 ) * ( 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 = SLAMCH( 'SAFEMIN' )
BIGNUM = ONE / SMLNUM
SMIN = BIGNUM
SMAX = ZERO
T = ONE / SQRT( AVG )
BASE = SLAMCH( '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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