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.

**FURTHER** **DETAILS**

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)