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

SUBROUTINE DLASDQ(

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

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

DLASDQ 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 * P' (P' 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 P' * VT if desired.

The input matrix C is changed to Q' * 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.

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 = 'U' or 'u' B is upper bidiagonal. UPLO = 'L' or 'l' 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 = 'U' and (N+1)-by-N if UPLU = 'L'.

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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDVT, NCVT)

On entry, contains a matrix which on exit has been premultiplied by P', 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDC, NCC)

On entry, contains an N-by-NCC matrix which on exit has been premultiplied by Q' 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) DOUBLE PRECISION 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.

Based on contributions by

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