--- rpl/lapack/lapack/zsyequb.f 2010/08/07 13:22:44 1.2
+++ rpl/lapack/lapack/zsyequb.f 2023/08/07 08:39:38 1.15
@@ -1,15 +1,139 @@
- SUBROUTINE ZSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
+*> \brief \b ZSYEQUB
*
-* -- LAPACK routine (version 3.2.2) --
-* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
-* -- Jason Riedy of Univ. of California Berkeley. --
-* -- June 2010 --
+* =========== DOCUMENTATION ===========
*
-* -- LAPACK is a software package provided by Univ. of Tennessee, --
-* -- Univ. of California Berkeley and NAG Ltd. --
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZSYEQUB + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZSYEQUB( 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
+*>
+*> ZSYEQUB computes row and column scalings intended to equilibrate a
+*> symmetric 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 symmetric 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 complex16SYcomputational
+*
+*> \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 ZSYEQUB( 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..--
*
- IMPLICIT NONE
-* ..
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION AMAX, SCOND
@@ -20,66 +144,6 @@
DOUBLE PRECISION S( * )
* ..
*
-* Purpose
-* =======
-*
-* 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.
-*
-* 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*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.
-*
-* LDA (input) INTEGER
-* The leading dimension of the array A. LDA >= max(1,N).
-*
-* S (output) DOUBLE PRECISION array, dimension (N)
-* If INFO = 0, S contains the scale factors for A.
-*
-* SCOND (output) 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.
-*
-* AMAX (output) 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.
-*
-* WORK (workspace) COMPLEX*16 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 ..
@@ -101,7 +165,7 @@
EXTERNAL DLAMCH, LSAME
* ..
* .. External Subroutines ..
- EXTERNAL ZLASSQ
+ EXTERNAL ZLASSQ, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
@@ -109,7 +173,7 @@
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
-* Statement Function Definitions
+* .. Statement Function Definitions ..
CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
* ..
* .. Executable Statements ..
@@ -118,15 +182,15 @@
*
INFO = 0
IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
- INFO = -1
+ INFO = -1
ELSE IF ( N .LT. 0 ) THEN
- INFO = -2
+ INFO = -2
ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
- INFO = -4
+ INFO = -4
END IF
IF ( INFO .NE. 0 ) THEN
- CALL XERBLA( 'ZSYEQUB', -INFO )
- RETURN
+ CALL XERBLA( 'ZSYEQUB', -INFO )
+ RETURN
END IF
UP = LSAME( UPLO, 'U' )
@@ -135,12 +199,12 @@
* Quick return if possible.
*
IF ( N .EQ. 0 ) THEN
- SCOND = ONE
- RETURN
+ SCOND = ONE
+ RETURN
END IF
DO I = 1, N
- S( I ) = ZERO
+ S( I ) = ZERO
END DO
AMAX = ZERO
@@ -151,7 +215,7 @@
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) ) )
+ S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
END DO
ELSE
@@ -160,102 +224,101 @@
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 ) ) )
+ 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 )
+ S( J ) = 1.0D0 / 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 = 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 ) * ( 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
+ 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
- DO J = I+1,N
- T = CABS1( A( J, I ) )
- U = U + S( J )*T
- WORK( J ) = WORK( J ) + D*T
+ 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 = AVG + ( U + WORK( I ) ) * D / N
- S( I ) = SI
- END DO
+ END IF
+
+* avg = s^T beta / n
+ AVG = 0.0D0
+ DO I = 1, N
+ AVG = AVG + S( I ) * DBLE( 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
@@ -268,9 +331,9 @@
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 ) )
+ 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 )
*