
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*XB, where A is NbyN upper
bidiagonal, and X and B are NbyNRHS
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*XB, where A is NbyN upper
bidiagonal, and X and B are NbyNRHS. 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 con
ceivably 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 (N1)
Contains the superdiagonal 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 nega
tive, 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 ith argument had an illegal value.
> 0: The algorithm failed to compute an singular value while working on the subma
trix lying in rows and columns INFO/(N+1) through MOD(INFO,N+1).
FURTHER DETAILS
Based on contributions by
Ming Gu and RenCang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
LAPACK version 3.0 15 June 2000 DLALSD(l) 
