--- rpl/lapack/lapack/dsyequb.f 2016/08/27 15:34:39 1.10
+++ rpl/lapack/lapack/dsyequb.f 2017/06/17 10:54:04 1.11
@@ -2,24 +2,24 @@
*
* =========== DOCUMENTATION ===========
*
-* Online html documentation available at
-* http://www.netlib.org/lapack/explore-html/
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
-*> Download DSYEQUB + dependencies
-*>
-*> [TGZ]
-*>
-*> [ZIP]
-*>
+*> Download DSYEQUB + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
*> [TXT]
-*> \endhtmlonly
+*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
-*
+*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, N
* DOUBLE PRECISION AMAX, SCOND
@@ -28,7 +28,7 @@
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
* ..
-*
+*
*
*> \par Purpose:
* =============
@@ -36,12 +36,11 @@
*> \verbatim
*>
*> DSYEQUB 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
+*> 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
*
@@ -51,30 +50,27 @@
*> \param[in] UPLO
*> \verbatim
*> UPLO is 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.
+*> = '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.
+*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION 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.
+*> 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).
+*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] S
@@ -87,21 +83,21 @@
*> \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
+*> 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.
+*> 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 DOUBLE PRECISION array, dimension (3*N)
+*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
@@ -115,12 +111,12 @@
* Authors:
* ========
*
-*> \author Univ. of Tennessee
-*> \author Univ. of California Berkeley
-*> \author Univ. of Colorado Denver
-*> \author NAG Ltd.
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
*
-*> \date November 2011
+*> \date December 2016
*
*> \ingroup doubleSYcomputational
*
@@ -130,15 +126,15 @@
*> 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://ruready.utah.edu/archive/papers/bin.pdf
+*> Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
*>
* =====================================================================
SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
*
-* -- LAPACK computational routine (version 3.4.0) --
+* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2011
+* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
@@ -153,7 +149,7 @@
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
- PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
+ PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
INTEGER MAX_ITER
PARAMETER ( MAX_ITER = 100 )
* ..
@@ -176,19 +172,19 @@
* ..
* .. Executable Statements ..
*
-* Test input parameters.
+* Test the input parameters.
*
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( 'DSYEQUB', -INFO )
- RETURN
+ CALL XERBLA( 'DSYEQUB', -INFO )
+ RETURN
END IF
UP = LSAME( UPLO, 'U' )
@@ -197,12 +193,12 @@
* 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
@@ -211,7 +207,7 @@
DO I = 1, J-1
S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
- AMAX = MAX( AMAX, ABS( A(I, J) ) )
+ AMAX = MAX( AMAX, ABS( A( I, J ) ) )
END DO
S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
AMAX = MAX( AMAX, ABS( A( J, J ) ) )
@@ -228,99 +224,95 @@
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)
+ 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 = ABS( A( I, J ) )
- WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
- WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
- END DO
- WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
- END DO
- ELSE
- DO J = 1, N
- WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
- DO I = J+1, N
- T = ABS( A( I, J ) )
- WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
- WORK( J ) = WORK( J ) + ABS( 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 DLASSQ( 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 = ABS( 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 = ABS( A( J, I ) )
- U = U + S( J )*T
- WORK( J ) = WORK( J ) + D*T
- END DO
- DO J = I+1,N
- T = ABS( A( I, J ) )
- U = U + S( J )*T
- WORK( J ) = WORK( J ) + D*T
- END DO
- ELSE
- DO J = 1, I
- T = ABS( 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 ) + ABS( A( I, J ) ) * S( J )
+ WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
+ END DO
+ WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
END DO
- DO J = I+1,N
- T = ABS( A( J, I ) )
- U = U + S( J )*T
- WORK( J ) = WORK( J ) + D*T
+ ELSE
+ DO J = 1, N
+ WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
+ DO I = J+1, N
+ WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
+ WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
+ END DO
END DO
- END IF
+ END IF
- AVG = AVG + ( U + WORK( I ) ) * D / N
- S( I ) = SI
+* avg = s^T beta / n
+ AVG = 0.0D0
+ DO I = 1, N
+ AVG = AVG + 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 DLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
+ STD = SCALE * SQRT( SUMSQ / N )
- END DO
+ IF ( STD .LT. TOL * AVG ) GOTO 999
+ DO I = 1, N
+ T = ABS( 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 = ABS( A( J, I ) )
+ U = U + S( J )*T
+ WORK( J ) = WORK( J ) + D*T
+ END DO
+ DO J = I+1,N
+ T = ABS( A( I, J ) )
+ U = U + S( J )*T
+ WORK( J ) = WORK( J ) + D*T
+ END DO
+ ELSE
+ DO J = 1, I
+ T = ABS( A( I, J ) )
+ U = U + S( J )*T
+ WORK( J ) = WORK( J ) + D*T
+ END DO
+ DO J = I+1,N
+ T = ABS( 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
@@ -329,13 +321,13 @@
BIGNUM = ONE / SMLNUM
SMIN = BIGNUM
SMAX = ZERO
- T = ONE / SQRT(AVG)
+ 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 ) )
+ 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 )
*