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 NRbyNR square matrix.
= 1: the lower block is an NRby(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 nondeflated 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 (NK) updated singular values (those which were deflated) sorted into increasing order.
 Z (output) DOUBLE PRECISION array, dimension(N)
 On exit Z contains the updating row vector in the secular equation.
 ALPHA (input) DOUBLE PRECISION
 Contains the diagonal element associated with the added row.
 BETA (input) DOUBLE PRECISION
 Contains the offdiagonal 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 (NK) updated left singular vectors (those which were deflated) in its last NK 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 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 (NK) updated right singular vectors (those which were deflated) in its last NK 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 (K1 singular values and one zero) in the secular equation.
 U2 (output) DOUBLE PRECISION array, dimension(LDU2,N)
 Contains a copy of the first K1 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 nonzero entries only at and above NL; the third contains nonzero 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 nonzeros only at and before NL +1; the third block contains nonzeros 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 Dvalues 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 nonzero entries only at and above NL, the second contains nonzero 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
subproblems 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 (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 : nonzero in the upper half only
2 : nonzero 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 Ith type columns.  INFO (output) INTEGER

= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
FURTHER DETAILS
Based on contributions byMing Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA