CLALSD - 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

SUBROUTINE CLALSD(

UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, RWORK, IWORK, INFO )

CHARACTER UPLO

INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ

REAL RCOND

INTEGER IWORK( * )

REAL D( * ), E( * ), RWORK( * )

COMPLEX B( LDB, * ), WORK( * )

CLALSD 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.

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) REAL 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) REAL array, dimension (N-1)

Contains the super-diagonal entries of the bidiagonal matrix. On exit, E has been destroyed.

B (input/output) COMPLEX 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) REAL

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) COMPLEX array, dimension at least

(N * NRHS).

RWORK (workspace) REAL 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 )/(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).

Based on contributions by

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Osni Marques, LBNL/NERSC, USA