dlasd2.f

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      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 )
*
*  -- LAPACK auxiliary routine (instrumented to count ops, version 3.0) --
*     Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
*     Courant Institute, NAG Ltd., and Rice University
*     October 31, 1999
*
*     .. 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( * )
*     ..
*     .. Common block to return operation count ..
      COMMON             / LATIME / OPS, ITCNT
*     ..
*     .. Scalars in Common ..
      DOUBLE PRECISION   ITCNT, OPS
*     ..
*
*  Purpose
*  =======
*
*  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.
*
*  Arguments
*  =========
*
*  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 N = NL + NR + 1 rows and
*         M = N + SQRE >= N columns.
*
*  K      (output) INTEGER
*         Contains the dimension of the non-deflated matrix,
*         This is the order of the related secular equation. 1 <= K <=N.
*
*  D      (input/output) 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.
*
*  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.
*
*  U      (input/output) 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.
*
*  LDU    (input) INTEGER
*         The leading dimension of the array U.  LDU >= N.
*
*  Z      (output) DOUBLE PRECISION array, dimension(N)
*         On exit Z contains the updating row vector in the secular
*         equation.
*
*  DSIGMA (output) DOUBLE PRECISION array, dimension (N)
*         Contains a copy of the diagonal elements (K-1 singular values
*         and one zero) in the secular equation.
*
*  U2     (output) 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.
*
*  LDU2   (input) INTEGER
*         The leading dimension of the array U2.  LDU2 >= N.
*
*  VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
*         On entry VT' 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' 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.
*
*  LDVT   (input) INTEGER
*         The leading dimension of the array VT.  LDVT >= M.
*
*  VT2    (output) DOUBLE PRECISION array, dimension(LDVT2,N)
*         VT2' 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.
*
*  LDVT2  (input) INTEGER
*         The leading dimension of the array VT2.  LDVT2 >= M.
*
*  IDXP   (workspace) 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.
*
*  IDX    (workspace) INTEGER array, dimension(N)
*         This will contain the permutation used to sort the contents of
*         D into ascending order.
*
*  IDXC   (output) 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.
*
*  COLTYP (workspace/output) 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.
*
*  IDXQ   (input) 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.
*
*  INFO   (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Ming Gu and Huan Ren, Computer Science Division, University of
*     California at Berkeley, USA
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO, EIGHT
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0,
     $                   EIGHT = 8.0D0 )
*     ..
*     .. 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          DBLE, ABS, MAX
*     ..
*     .. 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
*
      IF( LDU.LT.N ) THEN
         INFO = -10
      ELSE IF( LDVT.LT.M ) THEN
         INFO = -12
      ELSE IF( LDU2.LT.N ) THEN
         INFO = -15
      ELSE IF( LDVT2.LT.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.
*
      OPS = OPS + DBLE( 1 + NL )
      Z1 = ALPHA*VT( NLP1, NLP1 )
      Z( 1 ) = Z1
      DO 10 I = NL, 1, -1
         Z( I+1 ) = ALPHA*VT( I, NLP1 )
         D( I+1 ) = D( I )
         IDXQ( I+1 ) = IDXQ( I ) + 1
   10 CONTINUE
*
*     Generate the second part of the vector Z.
*
      OPS = OPS + DBLE( M-NLP2+1 )
      DO 20 I = NLP2, M
         Z( I ) = BETA*VT( I, NLP2 )
   20 CONTINUE
*
*     Initialize some reference arrays.
*
      DO 30 I = 2, NLP1
         COLTYP( I ) = 1
   30 CONTINUE
      DO 40 I = NLP2, N
         COLTYP( I ) = 2
   40 CONTINUE
*
*     Sort the singular values into increasing order
*
      DO 50 I = NLP2, N
         IDXQ( I ) = IDXQ( I ) + NLP1
   50 CONTINUE
*