Provided by: liblapack-doc_3.3.1-1_all #### NAME

```       LAPACK-3 - merges the two sets of singular values together into a single sorted set

```

#### SYNOPSIS

```       SUBROUTINE SLASD2( 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

REAL           ALPHA, BETA

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

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

```

#### PURPOSE

```       SLASD2  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.
SLASD2 is called from SLASD1.

```

#### 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) REAL 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      (output) REAL array, dimension (N)
On exit Z contains the updating row vector in the secular
equation.

ALPHA  (input) REAL
Contains the diagonal element associated with the added row.

BETA   (input) REAL
Contains the off-diagonal element associated with the added
row.

U      (input/output) REAL 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) REAL 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   (input) INTEGER
The leading dimension of the array VT.  LDVT >= M.
DSIGMA (output) REAL array, dimension (N)
Contains a copy of the diagonal elements (K-1 singular values
and one zero) in the secular equation.

U2     (output) REAL array, dimension (LDU2,N)
Contains a copy of the first K-1 left singular vectors which
will be used by SLASD3 in a matrix multiply (SGEMM) 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.

VT2    (output) REAL array, dimension (LDVT2,N)
VT2**T contains a copy of the first K right singular vectors
which will be used by SLASD3 in a matrix multiply (SGEMM) 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.

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
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 : 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.

```

#### FURTHERDETAILS

```        Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA

LAPACK auxiliary routine (version 3.2)     April 2011                            SLASD2(3lapack)
```