dlasd2_deflate.f

*> \brief \b DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
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*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT,
*                          LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX,
*                          IDXC, IDXQ, COLTYP, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
*       DOUBLE PRECISION   ALPHA, BETA
*       ..
*       .. Array Arguments ..
*       INTEGER            COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ),
*      $                   IDXQ( * )
*       DOUBLE PRECISION   D( * ), DSIGMA( * ), U( LDU, * ),
*      $                   U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
*      $                   Z( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DLASD2 merges the two sets of singular values together into a single
*> sorted set.  Then it tries to deflate the size of the problem.
*> There are two ways in which deflation can occur:  when two or more
*> singular values are close together or if there is a tiny entry in the
*> Z vector.  For each such occurrence the order of the related secular
*> equation problem is reduced by one.
*>
*> DLASD2 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[out] K
*> \verbatim
*>          K is INTEGER
*>         Contains the dimension of the non-deflated matrix,
*>         This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension(N)
*>         On entry D contains the singular values of the two submatrices
*>         to be combined.  On exit D contains the trailing (N-K) updated
*>         singular values (those which were deflated) sorted into
*>         increasing order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension(N)
*>         On exit Z contains the updating row vector in the secular
*>         equation.
*> \endverbatim
*>
*> \param[in] ALPHA
*> \verbatim
*>          ALPHA is DOUBLE PRECISION
*>         Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION
*>         Contains the off-diagonal element associated with the added
*>         row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is DOUBLE PRECISION array, dimension(LDU,N)
*>         On entry U contains the left singular vectors of two
*>         submatrices in the two square blocks with corners at (1,1),
*>         (NL, NL), and (NL+2, NL+2), (N,N).
*>         On exit U contains the trailing (N-K) updated left singular
*>         vectors (those which were deflated) in its last N-K columns.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         The leading dimension of the array U.  LDU >= N.
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is DOUBLE PRECISION array, dimension(LDVT,M)
*>         On entry VT**T contains the right singular vectors of two
*>         submatrices in the two square blocks with corners at (1,1),
*>         (NL+1, NL+1), and (NL+2, NL+2), (M,M).
*>         On exit VT**T contains the trailing (N-K) updated right singular
*>         vectors (those which were deflated) in its last N-K columns.
*>         In case SQRE =1, the last row of VT spans the right null
*>         space.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         The leading dimension of the array VT.  LDVT >= M.
*> \endverbatim
*>
*> \param[out] DSIGMA
*> \verbatim
*>          DSIGMA is DOUBLE PRECISION array, dimension (N)
*>         Contains a copy of the diagonal elements (K-1 singular values
*>         and one zero) in the secular equation.
*> \endverbatim
*>
*> \param[out] U2
*> \verbatim
*>          U2 is DOUBLE PRECISION array, dimension(LDU2,N)
*>         Contains a copy of the first K-1 left singular vectors which
*>         will be used by DLASD3 in a matrix multiply (DGEMM) to solve
*>         for the new left singular vectors. U2 is arranged into four
*>         blocks. The first block contains a column with 1 at NL+1 and
*>         zero everywhere else; the second block contains non-zero
*>         entries only at and above NL; the third contains non-zero
*>         entries only below NL+1; and the fourth is dense.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*>          LDU2 is INTEGER
*>         The leading dimension of the array U2.  LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT2
*> \verbatim
*>          VT2 is DOUBLE PRECISION array, dimension(LDVT2,N)
*>         VT2**T contains a copy of the first K right singular vectors
*>         which will be used by DLASD3 in a matrix multiply (DGEMM) to
*>         solve for the new right singular vectors. VT2 is arranged into
*>         three blocks. The first block contains a row that corresponds
*>         to the special 0 diagonal element in SIGMA; the second block
*>         contains non-zeros only at and before NL +1; the third block
*>         contains non-zeros only at and after  NL +2.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*>          LDVT2 is INTEGER
*>         The leading dimension of the array VT2.  LDVT2 >= M.
*> \endverbatim
*>
*> \param[out] IDXP
*> \verbatim
*>          IDXP is INTEGER array, dimension(N)
*>         This will contain the permutation used to place deflated
*>         values of D at the end of the array. On output IDXP(2:K)
*>         points to the nondeflated D-values and IDXP(K+1:N)
*>         points to the deflated singular values.
*> \endverbatim
*>
*> \param[out] IDX
*> \verbatim
*>          IDX is INTEGER array, dimension(N)
*>         This will contain the permutation used to sort the contents of
*>         D into ascending order.
*> \endverbatim
*>
*> \param[out] IDXC
*> \verbatim
*>          IDXC is INTEGER array, dimension(N)
*>         This will contain the permutation used to arrange the columns
*>         of the deflated U matrix into three groups:  the first group
*>         contains non-zero entries only at and above NL, the second
*>         contains non-zero entries only below NL+2, and the third is
*>         dense.
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*>          IDXQ is INTEGER array, dimension(N)
*>         This contains the permutation which separately sorts the two
*>         sub-problems in D into ascending order.  Note that entries in
*>         the first hlaf of this permutation must first be moved one
*>         position backward; and entries in the second half
*>         must first have NL+1 added to their values.
*> \endverbatim
*>
*> \param[out] COLTYP
*> \verbatim
*>          COLTYP is INTEGER array, dimension(N)
*>         As workspace, this will contain a label which will indicate
*>         which of the following types a column in the U2 matrix or a
*>         row in the VT2 matrix is:
*>         1 : non-zero in the upper half only
*>         2 : non-zero in the lower half only
*>         3 : dense
*>         4 : deflated
*>
*>         On exit, it is an array of dimension 4, with COLTYP(I) being
*>         the dimension of the I-th type columns.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup dlasd2_deflate
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      subroutine dlasd2( nl, nr, sqre, k, D, Z, alpha, beta, U, ldu, VT,
     $                   ldvt, dsigma, U2, ldu2, VT2, ldvt2, idxp, idx,
     $                   idxc, idxq, coltype, info )

      !! lapack auxiliary routine
      !! lapack is a software package provided by univ. of tennessee,
      !! univ. of california berkeley, univ. of colorado denver and nag ltd..

      !! scalar arguments
      integer            info, k, ldu, ldu2, ldvt, ldvt2, nl, nr, sqre
      double precision   alpha, beta

      !! array arguments
      integer            coltype( * ), idx( * ), idxc( * ), idxp( * ),
     $                   idxq( * )
      double precision   d( * ), dsigma( * ), u( ldu, * ),
     $                   u2( ldu2, * ), vt( ldvt, * ), vt2( ldvt2, * ),
     $                   z( * )

      !! =====================================================================
      !! parameters
      double precision   zero, one, two, eight
      parameter( zero = 0.0d+0, one = 1.0d+0, two = 2.0d+0,
     $                   eight = 8.0d+0 )

      !! local arrays
      integer            ctot( 4 ), psm( 4 )

      !! local scalars
      integer            ct, i, idxi, idxj, idxjp, j, jp, jprev, k2, m,
     $                   n, nlp1, nlp2
      double precision   c, eps, hlftol, s, tau, tol, z1

      !! external functions
      double precision   dlamch, dlapy2
      external           dlamch, dlapy2

      !! external subroutines
      external           dcopy, dlacpy, dlamrg, dlaset, drot, xerbla

      !! intrinsic functions
      intrinsic          abs, max

      !! =====================================================================
      !! executable statements

      !! test the input parameters.
      info = 0

      if (nl < 1) then
         info = -1
      else if (nr < 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

      if (ldu < n) then
         info = -10
      else if (ldvt < m) then
         info = -12
      else if (ldu2 < n) then
         info = -15
      else if (ldvt2 < m) then
         info = -17
      end if
      if (info .ne. 0) then
         call xerbla( 'dlasd2', -info )
         return
      end if

      nlp1 = nl + 1
      nlp2 = nl + 2

      !! generate the first part of the vector Z; and move the singular
      !! values in the first part of D one position backward.
      z1 = alpha*vt( nl+1, nl+1 )
      z( 1 ) = z1
      do i = nl, 1, -1
         z( i+1 ) = alpha*vt( i, nl+1 )
         d( i+1 ) = d( i )
         idxq( i+1 ) = idxq( i ) + 1
      end do

      !! generate the second part of the vector Z.
      do i = nl+2, m
         z( i ) = beta*vt( i, nl+2 )
      end do

      !! initialize some reference arrays.
      do i = 2, nl+1
         coltype( i ) = 1
      end do
      do i = nl+2, n
         coltype( i ) = 2
      end do

      do i = nl+2, n
         idxq( i ) = idxq( i ) + nl+1
      end do

      !! dsigma, idxc, idxc, and the first column of U2
      !! are used as storage space.
      do i = 2, n
         dsigma( i ) =       d( idxq( i ) )
         u2( i, 1 )  =       z( idxq( i ) )
         idxc( i )   = coltype( idxq( i ) )
      end do

      !! Get permutation idx that sorts singular values into increasing order,
      !! merging the 2 sorted lists into 1 list.
      call dlamrg( nl, nr, dsigma( 2 ), 1, 1, idx( 2 ) )

      do i = 2, n
         idxi = 1 + idx( i )  !! +1
         d( i )       = dsigma( idxi )
         z( i )       =     u2( idxi, 1 )
         coltype( i ) =   idxc( idxi )
      end do

      !! calculate the allowable deflation tolerance
      eps = dlamch( 'epsilon' )
      tol = eight * eps * max( abs( d(n) ), abs( alpha ), abs( beta ) )

      !! There are 2 kinds of deflation -- first a value in the Z-vector
      !! is small, second two (or more) singular values are very close
      !! together (their difference is small).
      !!
      !! If the value in the Z-vector is small, we simply permute the
      !! array so that the corresponding singular value is moved to the
      !! end.
      !!
      !! If two values in the D-vector are close, we perform a two-sided
      !! rotation designed to make one of the corresponding Z-vector
      !! entries zero, and then permute the array so that the deflated
      !! singular value is moved to the end.
      !!
      !! If there are multiple singular values then the problem deflates.
      !! here the number of equal singular values are found.  as each equal
      !! singular value is found, an elementary reflector is computed to
      !! rotate the corresponding singular subspace so that the
      !! corresponding components of Z are zero in this new basis.
      k = 1
      k2 = n + 1
      !do j = 2, n
      !   if (abs( Z( j ) ) <= tol) then
      !      !! deflate due to small Z component.
      !      k2 = k2 - 1
      !      idxp( k2 ) = j
      !      coltype( j ) = 4
      !      if (j == n) then
      !         go to 120
      !      end if
      !   else
      !      jprev = j
      !      break  !! was: go to 90
      !   end if
      !end do
   90 !continue
      !j = jprev
      jprev = -1
      do j = 2 : n  !! was `100 continue`
  !100 continue
         !j = j + 1
         !if (j > n) then
         !   go to 110
         !end if
         if (abs( z( j ) ) <= tol) then
            !! deflate due to small Z component.
            k2 = k2 - 1
            idxp( k2 ) = j
            coltype( j ) = 4
         else if (jprev > 0) then
            if (abs( d( j ) - d( jprev ) ) <= tol) then
               !! singular values are close enough to allow deflation.
               s = z( jprev )
               c = z( j )

               !! find sqrt(a**2 + b**2) without overflow or
               !! destructive underflow.
               tau = dlapy2( c, s )
               c =  c / tau
               s = -s / tau
               z( j ) = tau
               z( jprev ) = zero

               !! apply back the givens rotation to the left and right
               !! singular vector matrices.
               idxjp = idxq( idx( jprev )+1 )
               idxj = idxq( idx( j )+1 )
               if (idxjp <= nl+1) then
                  idxjp = idxjp - 1
               end if
               if (idxj <= nl+1) then
                  idxj = idxj - 1
               end if
               call drot( n, u( 1, idxjp ), 1, u( 1, idxj ), 1, c, s )
               call drot( m, vt( idxjp, 1 ), ldvt, vt( idxj, 1 ), ldvt, c, s )
               if (coltype( j ) .ne. coltype( jprev )) then
                  coltype( j ) = 3
               end if
               coltype( jprev ) = 4
               k2 = k2 - 1
               idxp( k2 ) = jprev
               jprev = j
            else
               k = k + 1
               u2( k, 1 ) = z( jprev )
               dsigma( k ) = d( jprev )
               idxp( k ) = jprev
               jprev = j
            end if
         else
            !! todo: anything else to do here (jprev == 0)??
            jprev = j
         end if
      end do  !! was `go to 100`
      !go to 100
  110 continue

      !! record the last singular value.
      k = k + 1
      u2( k, 1 ) = z( jprev )
      dsigma( k ) = d( jprev )
      idxp( k ) = jprev

  120 continue

      !! count up the total number of the various types of columns, then
      !! form a permutation which positions the four column types into
      !! four groups of uniform structure (although one or more of these
      !! groups may be empty).
      do j = 1, 4
         ctot( j ) = 0
      end do
      do j = 2, n
         ct = coltype( j )
         ctot( ct ) = ctot( ct ) + 1
      end do

      !! psm(*) = position in submatrix (of types 1 through 4)
      psm( 1 ) = 2
      psm( 2 ) = 2 + ctot( 1 )
      psm( 3 ) = psm( 2 ) + ctot( 2 )
      psm( 4 ) = psm( 3 ) + ctot( 3 )

      !! fill out the idxc array so that the permutation which it induces
      !! will place all type-1 columns first, all type-2 columns next,
      !! then all type-3's, and finally all type-4's, starting from the
      !! second column. this applies similarly to the rows of VT.
      do j = 2, n
         jp = idxp( j )
         ct = coltype( jp )
         idxc( psm( ct ) ) = j
         psm( ct ) = psm( ct ) + 1
      end do

      !! sort the singular values and corresponding singular vectors into
      !! dsigma, U2, and VT2 respectively.  the singular values/vectors
      !! which were not deflated go into the first k slots of dsigma, U2,
      !! and VT2 respectively, while those which were deflated go into the
      !! last n - k slots, except that the first column/row will be treated
      !! separately.
      do j = 2, n
         jp = idxp( j )
         dsigma( j ) = d( jp )
         idxj = idxq( idx( idxp( idxc( j ) ) )+1 )
         if (idxj <= nl+1) then
            idxj = idxj - 1
         end if
         call dcopy( n, u( 1, idxj ), 1, u2( 1, j ), 1 )
         call dcopy( m, vt( idxj, 1 ), ldvt, vt2( j, 1 ), ldvt2 )
      end do

      !! determine dsigma(1), dsigma(2) and Z(1)
      dsigma( 1 ) = zero
      hlftol = tol / two
      if (abs( dsigma( 2 ) ) <= hlftol) then
         dsigma( 2 ) = hlftol
      end if
      if (m > n) then
         z( 1 ) = dlapy2( z1, z( m ) )
         if (z( 1 ) <= tol) then
            c = one
            s = zero
            z( 1 ) = tol
         else
            c = z1 / z( 1 )
            s = z( m ) / z( 1 )
         end if
      else
         if (abs( z1 ) <= tol) then
            z( 1 ) = tol
         else
            z( 1 ) = z1
         end if
      end if

      !! move the rest of the updating row to Z.
      call dcopy( k-1, u2( 2, 1 ), 1, z( 2 ), 1 )

      !! determine the first column of U2, the first row of VT2 and the
      !! last row of VT.
      call dlaset( 'a', n, 1, zero, zero, u2, ldu2 )
      u2( nl+1, 1 ) = one
      if (m > n) then
         do i = 1, nl+1
            vt( m, i ) = -s*vt( nl+1, i )
            vt2( 1, i ) = c*vt( nl+1, i )
         end do
         do i = nl+2, m
            vt2( 1, i ) = s*vt( m, i )
            vt( m, i ) = c*vt( m, i )
         end do
      else
         call dcopy( m, vt( nl+1, 1 ), ldvt, vt2( 1, 1 ), ldvt2 )
      end if
      if (m > n) then
         call dcopy( m, vt( m, 1 ), ldvt, vt2( m, 1 ), ldvt2 )
      end if

      !! the deflated singular values and their corresponding vectors go
      !! into the back of D, U, and V respectively.
      if (n > k) then
         call dcopy( n-k, dsigma( k+1 ), 1, d( k+1 ), 1 )
         call dlacpy( 'a', n, n-k, u2( 1, k+1 ), ldu2, u( 1, k+1 ),
     $                ldu )
         call dlacpy( 'a', n-k, m, vt2( k+1, 1 ), ldvt2, vt( k+1, 1 ),
     $                ldvt )
      end if

      !! copy ctot into coltype for referencing in dlasd3.
      do j = 1, 4
         coltype( j ) = ctot( j )
      end do

      return

      !! end of dlasd2
      end