dlasd6.f

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      SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
     $                   IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
     $                   LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
     $                   IWORK, INFO )
*
*  -- LAPACK auxiliary routine (instrumented to count ops, version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      INTEGER            GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
     $                   NR, SQRE
      DOUBLE PRECISION   ALPHA, BETA, C, S
*     ..
*     .. Array Arguments ..
      INTEGER            GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
     $                   PERM( * )
      DOUBLE PRECISION   D( * ), DIFL( * ), DIFR( * ),
     $                   GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
     $                   VF( * ), VL( * ), WORK( * ), Z( * )
*     ..
*     .. Common block to return operation count ..
      COMMON             / LATIME / OPS, ITCNT
*     ..
*     .. Scalars in Common ..
      DOUBLE PRECISION   ITCNT, OPS
*     ..
*
*  Purpose
*  =======
*
*  DLASD6 computes the SVD of an updated upper bidiagonal matrix B
*  obtained by merging two smaller ones by appending a row. This
*  routine is used only for the problem which requires all singular
*  values and optionally singular vector matrices in factored form.
*  B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
*  A related subroutine, DLASD1, handles the case in which all singular
*  values and singular vectors of the bidiagonal matrix are desired.
*
*  DLASD6 computes the SVD as follows:
*
*                ( D1(in)  0    0     0 )
*    B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
*                (   0     0   D2(in) 0 )
*
*      = U(out) * ( D(out) 0) * VT(out)
*
*  where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
*  with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*  elsewhere; and the entry b is empty if SQRE = 0.
*
*  The singular values of B can be computed using D1, D2, the first
*  components of all the right singular vectors of the lower block, and
*  the last components of all the right singular vectors of the upper
*  block. These components are stored and updated in VF and VL,
*  respectively, in DLASD6. Hence U and VT are not explicitly
*  referenced.
*
*  The singular values are stored in D. The algorithm consists of two
*  stages:
*
*        The first stage consists of deflating the size of the problem
*        when there are multiple singular values or if there is a zero
*        in the Z vector. For each such occurence the dimension of the
*        secular equation problem is reduced by one. This stage is
*        performed by the routine DLASD7.
*
*        The second stage consists of calculating the updated
*        singular values. This is done by finding the roots of the
*        secular equation via the routine DLASD4 (as called by DLASD8).
*        This routine also updates VF and VL and computes the distances
*        between the updated singular values and the old singular
*        values.
*
*  DLASD6 is called from DLASDA.
*
*  Arguments
*  =========
*
*  ICOMPQ (input) INTEGER
*         Specifies whether singular vectors are to be computed in
*         factored form:
*         = 0: Compute singular values only.
*         = 1: Compute singular vectors in factored form as well.
*
*  NL     (input) INTEGER
*         The row dimension of the upper block.  NL >= 1.
*
*  NR     (input) INTEGER
*         The row dimension of the lower block.  NR >= 1.
*
*  SQRE   (input) 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 row dimension N = NL + NR + 1,
*         and column dimension M = N + SQRE.
*
*  D      (input/output) DOUBLE PRECISION array, dimension ( NL+NR+1 ).
*         On entry D(1:NL,1:NL) contains the singular values of the
*         upper block, and D(NL+2:N) contains the singular values
*         of the lower block. On exit D(1:N) contains the singular
*         values of the modified matrix.
*
*  VF     (input/output) DOUBLE PRECISION array, dimension ( M )
*         On entry, VF(1:NL+1) contains the first components of all
*         right singular vectors of the upper block; and VF(NL+2:M)
*         contains the first components of all right singular vectors
*         of the lower block. On exit, VF contains the first components
*         of all right singular vectors of the bidiagonal matrix.
*
*  VL     (input/output) DOUBLE PRECISION array, dimension ( M )
*         On entry, VL(1:NL+1) contains the  last components of all
*         right singular vectors of the upper block; and VL(NL+2:M)
*         contains the last components of all right singular vectors of
*         the lower block. On exit, VL contains the last components of
*         all right singular vectors of the bidiagonal matrix.
*
*  ALPHA  (input) DOUBLE PRECISION
*         Contains the diagonal element associated with the added row.
*
*  BETA   (input) DOUBLE PRECISION
*         Contains the off-diagonal element associated with the added
*         row.
*
*  IDXQ   (output) INTEGER array, dimension ( N )
*         This contains the permutation which will reintegrate the
*         subproblem just solved back into sorted order, i.e.
*         D( IDXQ( I = 1, N ) ) will be in ascending order.
*
*  PERM   (output) INTEGER array, dimension ( N )
*         The permutations (from deflation and sorting) to be applied
*         to each block. Not referenced if ICOMPQ = 0.
*
*  GIVPTR (output) INTEGER
*         The number of Givens rotations which took place in this
*         subproblem. Not referenced if ICOMPQ = 0.
*
*  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
*         Each pair of numbers indicates a pair of columns to take place
*         in a Givens rotation. Not referenced if ICOMPQ = 0.
*
*  LDGCOL (input) INTEGER
*         leading dimension of GIVCOL, must be at least N.
*
*  GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*         Each number indicates the C or S value to be used in the
*         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
*
*  LDGNUM (input) INTEGER
*         The leading dimension of GIVNUM and POLES, must be at least N.
*
*  POLES  (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*         On exit, POLES(1,*) is an array containing the new singular
*         values obtained from solving the secular equation, and
*         POLES(2,*) is an array containing the poles in the secular
*         equation. Not referenced if ICOMPQ = 0.
*
*  DIFL   (output) DOUBLE PRECISION array, dimension ( N )
*         On exit, DIFL(I) is the distance between I-th updated
*         (undeflated) singular value and the I-th (undeflated) old
*         singular value.
*
*  DIFR   (output) DOUBLE PRECISION array,
*                  dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
*                  dimension ( N ) if ICOMPQ = 0.
*         On exit, DIFR(I, 1) is the distance between I-th updated
*         (undeflated) singular value and the I+1-th (undeflated) old
*         singular value.
*
*         If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
*         normalizing factors for the right singular vector matrix.
*
*         See DLASD8 for details on DIFL and DIFR.
*
*  Z      (output) DOUBLE PRECISION array, dimension ( M )
*         The first elements of this array contain the components
*         of the deflation-adjusted updating row vector.
*
*  K      (output) INTEGER
*         Contains the dimension of the non-deflated matrix,
*         This is the order of the related secular equation. 1 <= K <=N.
*
*  C      (output) DOUBLE PRECISION
*         C contains garbage if SQRE =0 and the C-value of a Givens
*         rotation related to the right null space if SQRE = 1.
*
*  S      (output) DOUBLE PRECISION
*         S contains garbage if SQRE =0 and the S-value of a Givens
*         rotation related to the right null space if SQRE = 1.
*
*  WORK   (workspace) DOUBLE PRECISION array, dimension ( 4 * M )
*
*  IWORK  (workspace) INTEGER array, dimension ( 3 * N )
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = 1, an 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, ZERO
      PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
     $                   N, N1, N2
      DOUBLE PRECISION   ORGNRM
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      N = NL + NR + 1
      M = N + SQRE
*
      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
         INFO = -1
      ELSE IF( NL.LT.1 ) THEN
         INFO = -2
      ELSE IF( NR.LT.1 ) THEN
         INFO = -3
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -4
      ELSE IF( LDGCOL.LT.N ) THEN
         INFO = -14
      ELSE IF( LDGNUM.LT.N ) THEN
         INFO = -16
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLASD6', -INFO )
         RETURN
      END IF
*
*     The following values are for bookkeeping purposes only.  They are
*     integer pointers which indicate the portion of the workspace
*     used by a particular array in DLASD7 and DLASD8.
*
      ISIGMA = 1
      IW = ISIGMA + N
      IVFW = IW + M
      IVLW = IVFW + M
*
      IDX = 1
      IDXC = IDX + N
      IDXP = IDXC + N
*
*     Scale.
*
      ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
      D( NL+1 ) = ZERO
      DO 10 I = 1, N
         IF( ABS( D( I ) ).GT.ORGNRM ) THEN
            ORGNRM = ABS( D( I ) )
         END IF
   10 CONTINUE
      OPS = OPS + DBLE( N + 2 )
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
      ALPHA = ALPHA / ORGNRM
      BETA = BETA / ORGNRM
*
*     Sort and Deflate singular values.
*
      CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
     $             WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
     $             WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
     $             PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
     $             INFO )
*
*     Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
*
      CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
     $             WORK( ISIGMA ), WORK( IW ), INFO )
*
*     Save the poles if ICOMPQ = 1.
*
      IF( ICOMPQ.EQ.1 ) THEN
         CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 )
         CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
      END IF
*
*     Unscale.
*
      OPS = OPS + DBLE( N )
      CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
*     Prepare the IDXQ sorting permutation.
*
      N1 = K
      N2 = N - K
      CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
*
      RETURN
*
*     End of DLASD6
*
      END