dlasd6.f

          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 (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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( * )
    *     ..
    *
    *  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/output) DOUBLE PRECISION
    *         Contains the diagonal element associated with the added row.
    *
    *  BETA   (input/output) 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 = 1.0D+0, ZERO = 0.0D+0 )
    *     ..
    *     .. 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          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
          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.
    *
          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