#include "ESMF_LapackBlas.inc" !> \brief \b DLASD8 ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLASD8 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd8.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd8.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd8.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, ! DSIGMA, WORK, INFO ) ! ! .. Scalar Arguments .. ! INTEGER ICOMPQ, INFO, K, LDDIFR ! .. ! .. Array Arguments .. ! DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ), ! $ DSIGMA( * ), VF( * ), VL( * ), WORK( * ), ! $ Z( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> 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. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] ICOMPQ !> \verbatim !> ICOMPQ is 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. !> \endverbatim !> !> \param[in] K !> \verbatim !> K is INTEGER !> The number of terms in the rational function to be solved !> by DLASD4. K >= 1. !> \endverbatim !> !> \param[out] D !> \verbatim !> D is DOUBLE PRECISION array, dimension ( K ) !> On output, D contains the updated singular values. !> \endverbatim !> !> \param[in,out] Z !> \verbatim !> Z is 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. !> \endverbatim !> !> \param[in,out] VF !> \verbatim !> VF is 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. !> \endverbatim !> !> \param[in,out] VL !> \verbatim !> VL is 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. !> \endverbatim !> !> \param[out] DIFL !> \verbatim !> DIFL is DOUBLE PRECISION array, dimension ( K ) !> On exit, DIFL(I) = D(I) - DSIGMA(I). !> \endverbatim !> !> \param[out] DIFR !> \verbatim !> DIFR is 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. !> \endverbatim !> !> \param[in] LDDIFR !> \verbatim !> LDDIFR is INTEGER !> The leading dimension of DIFR, must be at least K. !> \endverbatim !> !> \param[in,out] DSIGMA !> \verbatim !> DSIGMA is 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. !> \endverbatim !> !> \param[out] WORK !> \verbatim !> WORK is DOUBLE PRECISION array, dimension at least 3 * K !> \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 = 1, a singular value did not converge !> \endverbatim ! ! Authors: ! ======== ! !> \author Univ. of Tennessee !> \author Univ. of California Berkeley !> \author Univ. of Colorado Denver !> \author NAG Ltd. ! !> \date November 2011 ! !> \ingroup auxOTHERauxiliary ! !> \par Contributors: ! ================== !> !> Ming Gu and Huan Ren, Computer Science Division, University of !> California at Berkeley, USA !> ! ===================================================================== SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, & & DSIGMA, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.4.0) -- ! -- 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 ICOMPQ, INFO, K, LDDIFR ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ), & & DSIGMA( * ), VF( * ), VL( * ), WORK( * ), & & Z( * ) ! .. ! ! ===================================================================== ! ! .. 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 CALL XERBLA( 'DLASD4', -INFO ) 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