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.

          NL is INTEGER
         The row dimension of the upper block.  NL >= 1.

          NR is INTEGER
         The row dimension of the lower block.  NR >= 1.

          SQRE is 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 is INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

          D is 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.

          Z is DOUBLE PRECISION array, dimension(N)
         On exit Z contains the updating row vector in the secular
         equation.

          ALPHA is DOUBLE PRECISION
         Contains the diagonal element associated with the added row.

          BETA is DOUBLE PRECISION
         Contains the off-diagonal element associated with the added
         row.

          U is 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 is INTEGER
         The leading dimension of the array U.  LDU >= N.

          VT is DOUBLE PRECISION array, dimension(LDVT,M)
         On entry VT**T 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**T 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 is INTEGER
         The leading dimension of the array VT.  LDVT >= M.

          DSIGMA is DOUBLE PRECISION array, dimension (N)
         Contains a copy of the diagonal elements (K-1 singular values
         and one zero) in the secular equation.

          U2 is 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 is INTEGER
         The leading dimension of the array U2.  LDU2 >= N.

          VT2 is DOUBLE PRECISION array, dimension(LDVT2,N)
         VT2**T 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 is INTEGER
         The leading dimension of the array VT2.  LDVT2 >= M.

          IDXP is 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 is INTEGER array dimension(N)
         This will contain the permutation used to sort the contents of
         D into ascending order.

          IDXC is 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.

          IDXQ is 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.

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