dlasd2 man page on Scientific

```DLASD2(1)	    LAPACK auxiliary routine (version 3.2)	     DLASD2(1)

NAME
DLASD2  - merges 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 (out‐
put) 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.

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

LAPACK auxiliary routine (versioNovember 2008			     DLASD2(1)
```
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