====================================================================
dlasd2 examples
====================================================================

====================================================================
Man Page Details
====================================================================

NAME
       DLASD2  - the two sets of singular values together into a single sorted
       set

SYNOPSIS
       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 )

           INTEGER        INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE

           DOUBLE         PRECISION ALPHA, BETA

           INTEGER        COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), IDXQ( *
                          )

           DOUBLE         PRECISION  D(  *  ),  DSIGMA(  * ), U( LDU, * ), U2(
                          LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )

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 increas-
              ing order.

       Z      (output) DOUBLE PRECISION array, dimension(N)
              On exit Z contains the updating row vector in the secular  equa-
              tion.

       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.

       VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
              On entry VT' contains the right singular vectors of  two  subma-
              trices  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 trail-
              ing  (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.

              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 every-
              where  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.

       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 con-
              tains non-zero entries only at and above NL, the second contains
              non-zero entries only below NL+2, and the third is dense.

       IDXQ   (input/output) 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 posi-
              tion backward; and entries in the second half  must  first  have
              NL+1 added to their values.

              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.

       INFO   (output) INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

