zlalsd(3)

NAME

ZLALSD - use the singular value decomposition of A to

solve the least squares problem of finding X to minimize the Eu

clidean 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  ZLALSD(  UPLO,  SMLSIZ, N, NRHS, D, E, B, LDB,
RCOND, RANK, WORK, RWORK, IWORK, INFO )
    CHARACTER      UPLO
    INTEGER        INFO, LDB, N, NRHS, RANK, SMLSIZ
    DOUBLE         PRECISION RCOND
    INTEGER        IWORK( * )
    DOUBLE         PRECISION D( * ), E( * ), RWORK( * )
    COMPLEX*16     B( LDB, * ), WORK( * )

PURPOSE

ZLALSD 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 sin

gular 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 sub

problems 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 bidi

agonal matrix. On exit, if INFO = 0, D contains its singular val

ues.

E (input) DOUBLE PRECISION array, dimension (N-1)

Contains the super-diagonal entries of the bidiago

nal matrix. On exit, E has been destroyed.

B (input/output) COMPLEX*16 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 subpro

gram. 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 sin

gular 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) COMPLEX*16 array, dimension at least

(N * NRHS).

RWORK (workspace) DOUBLE PRECISION array, dimension at

least

(9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +

(SMLSIZ+1)**2), where NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SML

SIZ+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 ille

gal 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

ZLALSD(3)