dlasd2 (l) - Linux Manuals

dlasd2: merges the two sets of singular values together into a single sorted set

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 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 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 VTaq 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 VTaq 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) 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 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) DOUBLE PRECISION array, dimension(LDVT2,N)
VT2aq 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 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

Pages related to dlasd2

  • dlasd2 (3)
  • dlasd0 (l) - a divide and conquer approach, DLASD0 computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE
  • dlasd1 (l) - computes the SVD of an upper bidiagonal N-by-M matrix B,
  • dlasd3 (l) - finds all the square roots of the roots of the secular equation, as defined by the values in D and Z
  • dlasd4 (l) - subroutine compute the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0
  • dlasd5 (l) - subroutine compute the square root of the I-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) * diag( D ) + RHO The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j
  • dlasd6 (l) - computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row
  • dlasd7 (l) - merges the two sets of singular values together into a single sorted set
  • dlasd8 (l) - finds the square roots of the roots of the secular equation,
  • dlasd9 (l) - find the square roots of the roots of the secular equation,
  • dlasda (l) - a divide and conquer approach, DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix B with diagonal D and offdiagonal E, where M = N + SQRE