Chimie Théorique » Opt'n Path

      SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,

     $                   DSIGMA, WORK, INFO )

*

*  -- LAPACK auxiliary 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..--

*     June 2010

*

*     .. Scalar Arguments ..

      INTEGER            ICOMPQ, INFO, K, LDDIFR

*     ..

*     .. Array Arguments ..

      DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( LDDIFR, * ),

     $                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),

     $                   Z( * )

*     ..

*

*  Purpose

*  =======

*

*  DLASD8 finds the square roots of the roots of the secular equation,

*  as defined by the values in DSIGMA and Z. It makes the appropriate

*  calls to DLASD4, and stores, for each  element in D, the distance

*  to its two nearest poles (elements in DSIGMA). It also updates

*  the arrays VF and VL, the first and last components of all the

*  right singular vectors of the original bidiagonal matrix.

*

*  DLASD8 is called from DLASD6.

*

*  Arguments

*  =========

*

*  ICOMPQ  (input) INTEGER

*          Specifies whether singular vectors are to be computed in

*          factored form in the calling routine:

*          = 0: Compute singular values only.

*          = 1: Compute singular vectors in factored form as well.

*

*  K       (input) INTEGER

*          The number of terms in the rational function to be solved

*          by DLASD4.  K >= 1.

*

*  D       (output) DOUBLE PRECISION array, dimension ( K )

*          On output, D contains the updated singular values.

*

*  Z       (input/output) DOUBLE PRECISION array, dimension ( K )

*          On entry, the first K elements of this array contain the

*          components of the deflation-adjusted updating row vector.

*          On exit, Z is updated.

*

*  VF      (input/output) DOUBLE PRECISION array, dimension ( K )

*          On entry, VF contains  information passed through DBEDE8.

*          On exit, VF contains the first K components of the first

*          components of all right singular vectors of the bidiagonal

*          matrix.

*

*  VL      (input/output) DOUBLE PRECISION array, dimension ( K )

*          On entry, VL contains  information passed through DBEDE8.

*          On exit, VL contains the first K components of the last

*          components of all right singular vectors of the bidiagonal

*          matrix.

*

*  DIFL    (output) DOUBLE PRECISION array, dimension ( K )

*          On exit, DIFL(I) = D(I) - DSIGMA(I).

*

*  DIFR    (output) DOUBLE PRECISION array,

*                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and

*                   dimension ( K ) if ICOMPQ = 0.

*          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not

*          defined and will not be referenced.

*

*          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the

*          normalizing factors for the right singular vector matrix.

*

*  LDDIFR  (input) INTEGER

*          The leading dimension of DIFR, must be at least K.

*

*  DSIGMA  (input/output) DOUBLE PRECISION array, dimension ( K )

*          On entry, the first K elements of this array contain the old

*          roots of the deflated updating problem.  These are the poles

*          of the secular equation.

*          On exit, the elements of DSIGMA may be very slightly altered

*          in value.

*

*  WORK    (workspace) DOUBLE PRECISION array, dimension at least 3 * K

*

*  INFO    (output) INTEGER

*          = 0:  successful exit.

*          < 0:  if INFO = -i, the i-th argument had an illegal value.

*          > 0:  if INFO = 1, a singular value did not converge

*

*  Further Details

*  ===============

*

*  Based on contributions by

*     Ming Gu and Huan Ren, Computer Science Division, University of

*     California at Berkeley, USA

*

*  =====================================================================

*

*     .. Parameters ..

      DOUBLE PRECISION   ONE

      PARAMETER          ( ONE = 1.0D+0 )

*     ..

*     .. Local Scalars ..

      INTEGER            I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J

      DOUBLE PRECISION   DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP

*     ..

*     .. External Subroutines ..

      EXTERNAL           DCOPY, DLASCL, DLASD4, DLASET, XERBLA

*     ..

*     .. External Functions ..

      DOUBLE PRECISION   DDOT, DLAMC3, DNRM2

      EXTERNAL           DDOT, DLAMC3, DNRM2

*     ..

*     .. Intrinsic Functions ..

      INTRINSIC          ABS, SIGN, SQRT

*     ..

*     .. Executable Statements ..

*

*     Test the input parameters.

*

      INFO = 0

*

      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN

         INFO = -1

      ELSE IF( K.LT.1 ) THEN

         INFO = -2

      ELSE IF( LDDIFR.LT.K ) THEN

         INFO = -9

      END IF

      IF( INFO.NE.0 ) THEN

         CALL XERBLA( 'DLASD8', -INFO )

         RETURN

      END IF

*

*     Quick return if possible

*

      IF( K.EQ.1 ) THEN

         D( 1 ) = ABS( Z( 1 ) )

         DIFL( 1 ) = D( 1 )

         IF( ICOMPQ.EQ.1 ) THEN

            DIFL( 2 ) = ONE

            DIFR( 1, 2 ) = ONE

         END IF

         RETURN

      END IF

*

*     Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can

*     be computed with high relative accuracy (barring over/underflow).

*     This is a problem on machines without a guard digit in

*     add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).

*     The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),

*     which on any of these machines zeros out the bottommost

*     bit of DSIGMA(I) if it is 1; this makes the subsequent

*     subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation

*     occurs. On binary machines with a guard digit (almost all

*     machines) it does not change DSIGMA(I) at all. On hexadecimal

*     and decimal machines with a guard digit, it slightly

*     changes the bottommost bits of DSIGMA(I). It does not account

*     for hexadecimal or decimal machines without guard digits

*     (we know of none). We use a subroutine call to compute

*     2*DLAMBDA(I) to prevent optimizing compilers from eliminating

*     this code.

*

      DO 10 I = 1, K

         DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )

   10 CONTINUE

*

*     Book keeping.

*

      IWK1 = 1

      IWK2 = IWK1 + K

      IWK3 = IWK2 + K

      IWK2I = IWK2 - 1

      IWK3I = IWK3 - 1

*

*     Normalize Z.

*

      RHO = DNRM2( K, Z, 1 )

      CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )

      RHO = RHO*RHO

*

*     Initialize WORK(IWK3).

*

      CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K )

*

*     Compute the updated singular values, the arrays DIFL, DIFR,

*     and the updated Z.

*

      DO 40 J = 1, K

         CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ),

     $                WORK( IWK2 ), INFO )

*

*        If the root finder fails, the computation is terminated.

*

         IF( INFO.NE.0 ) THEN

            RETURN

         END IF

         WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )

         DIFL( J ) = -WORK( J )

         DIFR( J, 1 ) = -WORK( J+1 )

         DO 20 I = 1, J - 1

            WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*

     $                        WORK( IWK2I+I ) / ( DSIGMA( I )-

     $                        DSIGMA( J ) ) / ( DSIGMA( I )+

     $                        DSIGMA( J ) )

   20    CONTINUE

         DO 30 I = J + 1, K

            WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*

     $                        WORK( IWK2I+I ) / ( DSIGMA( I )-

     $                        DSIGMA( J ) ) / ( DSIGMA( I )+

     $                        DSIGMA( J ) )

   30    CONTINUE

   40 CONTINUE

*

*     Compute updated Z.

*

      DO 50 I = 1, K

         Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )

   50 CONTINUE

*

*     Update VF and VL.

*

      DO 80 J = 1, K

         DIFLJ = DIFL( J )

         DJ = D( J )

         DSIGJ = -DSIGMA( J )

         IF( J.LT.K ) THEN

            DIFRJ = -DIFR( J, 1 )

            DSIGJP = -DSIGMA( J+1 )

         END IF

         WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )

         DO 60 I = 1, J - 1

            WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ )

     $                   / ( DSIGMA( I )+DJ )

   60    CONTINUE

         DO 70 I = J + 1, K

            WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ )

     $                   / ( DSIGMA( I )+DJ )

   70    CONTINUE

         TEMP = DNRM2( K, WORK, 1 )

         WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP

         WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP

         IF( ICOMPQ.EQ.1 ) THEN

            DIFR( J, 2 ) = TEMP

         END IF

   80 CONTINUE

*

      CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 )

      CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 )

*

      RETURN

*

*     End of DLASD8

*

      END