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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 (version 3.2.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     June 2010

*

*     .. 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, a 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