dlasd3.f

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*> \brief \b DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
*                          LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
*                          INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
*      $                   SQRE
*       ..
*       .. Array Arguments ..
*       INTEGER            CTOT( * ), IDXC( * )
*       DOUBLE PRECISION   D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
*      $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
*      $                   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD3 finds all the square roots of the roots of the secular
*> equation, as defined by the values in D and Z.  It makes the
*> appropriate calls to DLASD4 and then updates the singular
*> vectors by matrix multiplication.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*>
*> DLASD3 is called from DLASD1.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block.  NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block.  NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has N = NL + NR + 1 rows and
*>         M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>         The size of the secular equation, 1 =< K = < N.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension(K)
*>         On exit the square roots of the roots of the secular equation,
*>         in ascending order.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDQ,K)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= K.
*> \endverbatim
*>
*> \param[in,out] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION array, dimension(K)
*>         The first K elements of this array contain the old roots
*>         of the deflated updating problem.  These are the poles
*>         of the secular equation.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension (LDU, N)
*>         The last N - K columns of this matrix contain the deflated
*>         left singular vectors.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         The leading dimension of the array U.  LDU >= N.
*> \endverbatim
*>
*> \param[in] U2
*> \verbatim
*>          U2 is DOUBLE PRECISION array, dimension (LDU2, N)
*>         The first K columns of this matrix contain the non-deflated
*>         left singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*>          LDU2 is INTEGER
*>         The leading dimension of the array U2.  LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension (LDVT, M)
*>         The last M - K columns of VT**T contain the deflated
*>         right singular vectors.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         The leading dimension of the array VT.  LDVT >= N.
*> \endverbatim
*>
*> \param[in,out] VT2
*> \verbatim
*>          VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
*>         The first K columns of VT2**T contain the non-deflated
*>         right singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*>          LDVT2 is INTEGER
*>         The leading dimension of the array VT2.  LDVT2 >= N.
*> \endverbatim
*>
*> \param[in] IDXC
*> \verbatim
*>          IDXC is INTEGER array, dimension ( N )
*>         The permutation used to arrange the columns of U (and rows of
*>         VT) into three groups:  the first group contains non-zero
*>         entries only at and above (or before) NL +1; the second
*>         contains non-zero entries only at and below (or after) NL+2;
*>         and the third is dense. The first column of U and the row of
*>         VT are treated separately, however.
*>
*>         The rows of the singular vectors found by DLASD4
*>         must be likewise permuted before the matrix multiplies can
*>         take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*>          CTOT is INTEGER array, dimension ( 4 )
*>         A count of the total number of the various types of columns
*>         in U (or rows in VT), as described in IDXC. The fourth column
*>         type is any column which has been deflated.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (K)
*>         The first K elements of this array contain the components
*>         of the deflation-adjusted updating row vector.
*> \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 June 2017
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE dlasd3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
     $                   LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
     $                   INFO )
*
*  -- LAPACK auxiliary routine (version 3.7.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2017
*
*     .. Scalar Arguments ..
      INTEGER            INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
     $                   sqre
*     ..
*     .. Array Arguments ..
      INTEGER            CTOT( * ), IDXC( * )
      DOUBLE PRECISION   D( * ), DSIGMA( * ), Q( ldq, * ), U( ldu, * ),
     $                   u2( ldu2, * ), vt( ldvt, * ), vt2( ldvt2, * ),
     $                   z( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO, NEGONE
      parameter( one = 1.0d+0, zero = 0.0d+0,
     $                   negone = -1.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
      DOUBLE PRECISION   RHO, TEMP
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMC3, DNRM2
      EXTERNAL           dlamc3, dnrm2
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dgemm, dlacpy, dlascl, dlasd4, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, sign, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( nl.LT.1 ) THEN
         info = -1
      ELSE IF( nr.LT.1 ) THEN
         info = -2
      ELSE IF( ( sqre.NE.1 ) .AND. ( sqre.NE.0 ) ) THEN
         info = -3
      END IF
*
      n = nl + nr + 1
      m = n + sqre
      nlp1 = nl + 1
      nlp2 = nl + 2
*
      IF( ( k.LT.1 ) .OR. ( k.GT.n ) ) THEN
         info = -4
      ELSE IF( ldq.LT.k ) THEN
         info = -7
      ELSE IF( ldu.LT.n ) THEN
         info = -10
      ELSE IF( ldu2.LT.n ) THEN
         info = -12
      ELSE IF( ldvt.LT.m ) THEN
         info = -14
      ELSE IF( ldvt2.LT.m ) THEN
         info = -16
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DLASD3', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( k.EQ.1 ) THEN
         d( 1 ) = abs( z( 1 ) )
         CALL dcopy( m, vt2( 1, 1 ), ldvt2, vt( 1, 1 ), ldvt )
         IF( z( 1 ).GT.zero ) THEN
            CALL dcopy( n, u2( 1, 1 ), 1, u( 1, 1 ), 1 )
         ELSE
            DO 10 i = 1, n
               u( i, 1 ) = -u2( i, 1 )
   10       CONTINUE
         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*DSIGMA(I) to prevent optimizing compilers from eliminating
*     this code.
*
      DO 20 i = 1, k
         dsigma( i ) = dlamc3( dsigma( i ), dsigma( i ) ) - dsigma( i )
   20 CONTINUE
*
*     Keep a copy of Z.
*
      CALL dcopy( k, z, 1, q, 1 )
*
*     Normalize Z.
*
      rho = dnrm2( k, z, 1 )
      CALL dlascl( 'G', 0, 0, rho, one, k, 1, z, k, info )
      rho = rho*rho
*
*     Find the new singular values.
*
      DO 30 j = 1, k
         CALL dlasd4( k, j, dsigma, z, u( 1, j ), rho, d( j ),
     $                vt( 1, j ), info )
*
*        If the zero finder fails, report the convergence failure.
*
         IF( info.NE.0 ) THEN
            RETURN
         END IF
   30 CONTINUE
*
*     Compute updated Z.
*
      DO 60 i = 1, k
         z( i ) = u( i, k )*vt( i, k )
         DO 40 j = 1, i - 1
            z( i ) = z( i )*( u( i, j )*vt( i, j ) /
     $               ( dsigma( i )-dsigma( j ) ) /
     $               ( dsigma( i )+dsigma( j ) ) )
   40    CONTINUE
         DO 50 j = i, k - 1
            z( i ) = z( i )*( u( i, j )*vt( i, j ) /
     $               ( dsigma( i )-dsigma( j+1 ) ) /
     $               ( dsigma( i )+dsigma( j+1 ) ) )
   50    CONTINUE
         z( i ) = sign( sqrt( abs( z( i ) ) ), q( i, 1 ) )
   60 CONTINUE
*
*     Compute left singular vectors of the modified diagonal matrix,
*     and store related information for the right singular vectors.
*
      DO 90 i = 1, k
         vt( 1, i ) = z( 1 ) / u( 1, i ) / vt( 1, i )
         u( 1, i ) = negone
         DO 70 j = 2, k
            vt( j, i ) = z( j ) / u( j, i ) / vt( j, i )
            u( j, i ) = dsigma( j )*vt( j, i )
   70    CONTINUE
         temp = dnrm2( k, u( 1, i ), 1 )
         q( 1, i ) = u( 1, i ) / temp
         DO 80 j = 2, k
            jc = idxc( j )
            q( j, i ) = u( jc, i ) / temp
   80    CONTINUE
   90 CONTINUE
*
*     Update the left singular vector matrix.
*
      IF( k.EQ.2 ) THEN
         CALL dgemm( 'N', 'N', n, k, k, one, u2, ldu2, q, ldq, zero, u,
     $               ldu )
         GO TO 100
      END IF
      IF( ctot( 1 ).GT.0 ) THEN
         CALL dgemm( 'N', 'N', nl, k, ctot( 1 ), one, u2( 1, 2 ), ldu2,
     $               q( 2, 1 ), ldq, zero, u( 1, 1 ), ldu )
         IF( ctot( 3 ).GT.0 ) THEN
            ktemp = 2 + ctot( 1 ) + ctot( 2 )
            CALL dgemm( 'N', 'N', nl, k, ctot( 3 ), one, u2( 1, ktemp ),
     $                  ldu2, q( ktemp, 1 ), ldq, one, u( 1, 1 ), ldu )
         END IF
      ELSE IF( ctot( 3 ).GT.0 ) THEN
         ktemp = 2 + ctot( 1 ) + ctot( 2 )
         CALL dgemm( 'N', 'N', nl, k, ctot( 3 ), one, u2( 1, ktemp ),
     $               ldu2, q( ktemp, 1 ), ldq, zero, u( 1, 1 ), ldu )
      ELSE
         CALL dlacpy( 'F', nl, k, u2, ldu2, u, ldu )
      END IF
      CALL dcopy( k, q( 1, 1 ), ldq, u( nlp1, 1 ), ldu )
      ktemp = 2 + ctot( 1 )
      ctemp = ctot( 2 ) + ctot( 3 )
      CALL dgemm( 'N', 'N', nr, k, ctemp, one, u2( nlp2, ktemp ), ldu2,
     $            q( ktemp, 1 ), ldq, zero, u( nlp2, 1 ), ldu )
*
*     Generate the right singular vectors.
*
  100 CONTINUE
      DO 120 i = 1, k
         temp = dnrm2( k, vt( 1, i ), 1 )
         q( i, 1 ) = vt( 1, i ) / temp
         DO 110 j = 2, k
            jc = idxc( j )
            q( i, j ) = vt( jc, i ) / temp
  110    CONTINUE
  120 CONTINUE
*
*     Update the right singular vector matrix.
*
      IF( k.EQ.2 ) THEN
         CALL dgemm( 'N', 'N', k, m, k, one, q, ldq, vt2, ldvt2, zero,
     $               vt, ldvt )
         RETURN
      END IF
      ktemp = 1 + ctot( 1 )
      CALL dgemm( 'N', 'N', k, nlp1, ktemp, one, q( 1, 1 ), ldq,
     $            vt2( 1, 1 ), ldvt2, zero, vt( 1, 1 ), ldvt )
      ktemp = 2 + ctot( 1 ) + ctot( 2 )
      IF( ktemp.LE.ldvt2 )
     $   CALL dgemm( 'N', 'N', k, nlp1, ctot( 3 ), one, q( 1, ktemp ),
     $               ldq, vt2( ktemp, 1 ), ldvt2, one, vt( 1, 1 ),
     $               ldvt )
*
      ktemp = ctot( 1 ) + 1
      nrp1 = nr + sqre
      IF( ktemp.GT.1 ) THEN
         DO 130 i = 1, k
            q( i, ktemp ) = q( i, 1 )
  130    CONTINUE
         DO 140 i = nlp2, m
            vt2( ktemp, i ) = vt2( 1, i )
  140    CONTINUE
      END IF
      ctemp = 1 + ctot( 2 ) + ctot( 3 )
      CALL dgemm( 'N', 'N', k, nrp1, ctemp, one, q( 1, ktemp ), ldq,
     $            vt2( ktemp, nlp2 ), ldvt2, zero, vt( 1, nlp2 ), ldvt )
*
      RETURN
*
*     End of DLASD3
*
      END