slasd8.f

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*> \brief \b SLASD8
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
*                          DSIGMA, WORK, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            ICOMPQ, INFO, K, LDDIFR
*       ..
*       .. Array Arguments ..
*       REAL               D( * ), DIFL( * ), DIFR( LDDIFR, * ),
*      $                   DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
*      $                   Z( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLASD8 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 SLASD4, 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.
*>
*> SLASD8 is called from SLASD6.
*> \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 SLASD4.  K >= 1.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is REAL array, dimension ( K )
*>          On output, D contains the updated singular values.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is REAL 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 REAL 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 REAL 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 REAL array, dimension ( K )
*>          On exit, DIFL(I) = D(I) - DSIGMA(I).
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*>          DIFR is REAL 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 REAL 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 REAL 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 slasd8( 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 ..
      REAL               d( * ), difl( * ), difr( lddifr, * ),
     $                   dsigma( * ), vf( * ), vl( * ), work( * ),
     $                   z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               one
      parameter( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            i, iwk1, iwk2, iwk2i, iwk3, iwk3i, j
      REAL               diflj, difrj, dj, dsigj, dsigjp, rho, temp
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, slascl, slasd4, slaset, xerbla
*     ..
*     .. External Functions ..
      REAL               sdot, slamc3, snrm2
      EXTERNAL           sdot, slamc3, snrm2
*     ..
*     .. 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( 'SLASD8', -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 ) = slamc3( 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 = snrm2( k, z, 1 )
      CALL slascl( 'G', 0, 0, rho, one, k, 1, z, k, info )
      rho = rho*rho
*
*     Initialize WORK(IWK3).
*
      CALL slaset( '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 slasd4( 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( 'SLASD4', -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 ) / ( slamc3( dsigma( i ), dsigj )-diflj )
     $                   / ( dsigma( i )+dj )
   60    CONTINUE
         DO 70 i = j + 1, k
            work( i ) = z( i ) / ( slamc3( dsigma( i ), dsigjp )+difrj )
     $                   / ( dsigma( i )+dj )
   70    CONTINUE
         temp = snrm2( k, work, 1 )
         work( iwk2i+j ) = sdot( k, work, 1, vf, 1 ) / temp
         work( iwk3i+j ) = sdot( k, work, 1, vl, 1 ) / temp
         IF( icompq.EQ.1 ) THEN
            difr( j, 2 ) = temp
         END IF
   80 CONTINUE
*
      CALL scopy( k, work( iwk2 ), 1, vf, 1 )
      CALL scopy( k, work( iwk3 ), 1, vl, 1 )
*
      RETURN
*
*     End of SLASD8
*
      END