slasdq (l) - Linux Manuals

slasdq: computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired

NAME

SLASDQ - computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired

SYNOPSIS

SUBROUTINE SLASDQ(
UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO )

    
CHARACTER UPLO

    
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE

    
REAL C( LDC, * ), D( * ), E( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )

PURPOSE

SLASDQ computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired. Letting B denote the input bidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q * S * Paq (Paq denotes the transpose of P). The singular values S are overwritten on D.
The input matrix U is changed to U * Q if desired.
The input matrix VT is changed to Paq * VT if desired.
The input matrix C is changed to Qaq * C if desired.
See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm.

ARGUMENTS

UPLO (input) CHARACTER*1
On entry, UPLO specifies whether the input bidiagonal matrix is upper or lower bidiagonal, and wether it is square are not. UPLO = aqUaq or aquaq B is upper bidiagonal. UPLO = aqLaq or aqlaq B is lower bidiagonal.
SQRE (input) INTEGER
= 0: then the input matrix is N-by-N.
= 1: then the input matrix is N-by-(N+1) if UPLU = aqUaq and (N+1)-by-N if UPLU = aqLaq. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns.
N (input) INTEGER
On entry, N specifies the number of rows and columns in the matrix. N must be at least 0.
NCVT (input) INTEGER
On entry, NCVT specifies the number of columns of the matrix VT. NCVT must be at least 0.
NRU (input) INTEGER
On entry, NRU specifies the number of rows of the matrix U. NRU must be at least 0.
NCC (input) INTEGER
On entry, NCC specifies the number of columns of the matrix C. NCC must be at least 0.
D (input/output) REAL array, dimension (N)
On entry, D contains the diagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in ascending order.
E (input/output) REAL array.
dimension is (N-1) if SQRE = 0 and N if SQRE = 1. On entry, the entries of E contain the offdiagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, E will contain 0. If the algorithm does not converge, D and E will contain the diagonal and superdiagonal entries of a bidiagonal matrix orthogonally equivalent to the one given as input.
VT (input/output) REAL array, dimension (LDVT, NCVT)
On entry, contains a matrix which on exit has been premultiplied by Paq, dimension N-by-NCVT if SQRE = 0 and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0).
LDVT (input) INTEGER
On entry, LDVT specifies the leading dimension of VT as declared in the calling (sub) program. LDVT must be at least 1. If NCVT is nonzero LDVT must also be at least N.
U (input/output) REAL array, dimension (LDU, N)
On entry, contains a matrix which on exit has been postmultiplied by Q, dimension NRU-by-N if SQRE = 0 and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0).
LDU (input) INTEGER
On entry, LDU specifies the leading dimension of U as declared in the calling (sub) program. LDU must be at least max( 1, NRU ) .
C (input/output) REAL array, dimension (LDC, NCC)
On entry, contains an N-by-NCC matrix which on exit has been premultiplied by Qaq dimension N-by-NCC if SQRE = 0 and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0).
LDC (input) INTEGER
On entry, LDC specifies the leading dimension of C as declared in the calling (sub) program. LDC must be at least 1. If NCC is nonzero, LDC must also be at least N.
WORK (workspace) REAL array, dimension (4*N)
Workspace. Only referenced if one of NCVT, NRU, or NCC is nonzero, and if N is at least 2.
INFO (output) INTEGER
On exit, a value of 0 indicates a successful exit. If INFO < 0, argument number -INFO is illegal. If INFO > 0, the algorithm did not converge, and INFO specifies how many superdiagonals did not converge.

FURTHER DETAILS

Based on contributions by

Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA

Pages related to slasdq

  • slasdq (3)
  • slasd0 (l) - a divide and conquer approach, SLASD0 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
  • slasd1 (l) - computes the SVD of an upper bidiagonal N-by-M matrix B,
  • slasd2 (l) - merges the two sets of singular values together into a single sorted set
  • slasd3 (l) - finds all the square roots of the roots of the secular equation, as defined by the values in D and Z
  • slasd4 (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
  • slasd5 (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
  • slasd6 (l) - computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row
  • slasd7 (l) - merges the two sets of singular values together into a single sorted set
  • slasd8 (l) - finds the square roots of the roots of the secular equation,
  • slasd9 (l) - find the square roots of the roots of the secular equation,