DLALSD(l) ) DLALSD(l)NAME
DLALSD - use the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
by-NRHS
SYNOPSIS
SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK,
WORK, IWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
DOUBLE PRECISION RCOND
INTEGER IWORK( * )
DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * )
PURPOSE
DLALSD uses the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
by-NRHS. The solution X overwrites B. The singular values of A smaller
than RCOND times the largest singular value are treated as zero in
solving the least squares problem; in this case a minimum norm solution
is returned. The actual singular values are returned in D in ascending
order.
This code makes very mild assumptions about floating point arithmetic.
It will work on machines with a guard digit in add/subtract, or on
those binary machines without guard digits which subtract like the Cray
XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on
hexadecimal or decimal machines without guard digits, but we know of
none.
ARGUMENTS
UPLO (input) CHARACTER*1
= 'U': D and E define an upper bidiagonal matrix.
= 'L': D and E define a lower bidiagonal matrix.
SMLSIZ (input) INTEGER The maximum size of the subproblems at
the bottom of the computation tree.
N (input) INTEGER
The dimension of the bidiagonal matrix. N >= 0.
NRHS (input) INTEGER
The number of columns of B. NRHS must be at least 1.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry D contains the main diagonal of the bidiagonal matrix.
On exit, if INFO = 0, D contains its singular values.
E (input) DOUBLE PRECISION array, dimension (N-1)
Contains the super-diagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
On input, B contains the right hand sides of the least squares
problem. On output, B contains the solution X.
LDB (input) INTEGER
The leading dimension of B in the calling subprogram. LDB must
be at least max(1,N).
RCOND (input) DOUBLE PRECISION
The singular values of A less than or equal to RCOND times the
largest singular value are treated as zero in solving the least
squares problem. If RCOND is negative, machine precision is used
instead. For example, if diag(S)*X=B were the least squares
problem, where diag(S) is a diagonal matrix of singular values,
the solution would be X(i) = B(i) / S(i) if S(i) is greater than
RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
RCOND*max(S).
RANK (output) INTEGER
The number of singular values of A greater than RCOND times the
largest singular value.
WORK (workspace) DOUBLE PRECISION array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), where
NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
IWORK (workspace) INTEGER array, dimension at least
(3*N*NLVL + 11*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an singular value while
working on the submatrix lying in rows and columns INFO/(N+1)
through MOD(INFO,N+1).
FURTHER DETAILS
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
LAPACK version 3.0 15 June 2000 DLALSD(l)
