SGELSD - compute the minimum-norm solution to a real linear least squares problem

SUBROUTINE SGELSD(

M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO )

INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

REAL RCOND

INTEGER IWORK( * )

REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

SGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |)

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The problem is solved in three steps:

(1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS)

(2) Solve the BLS using a divide and conquer approach.

(3) Apply back all the Householder tranformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

The divide and conquer algorithm 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

M (input) INTEGER

The number of rows of A. M >= 0.

N (input) INTEGER

The number of columns of A. N >= 0.

NRHS (input) INTEGER

The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

A (input) REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, A has been destroyed.

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,M).

B (input/output) REAL array, dimension (LDB,NRHS)

On entry, the M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column.

LDB (input) INTEGER

The leading dimension of the array B. LDB >= max(1,max(M,N)).

S (output) REAL array, dimension (min(M,N))

The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).

RCOND (input) REAL

RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.

RANK (output) INTEGER

The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).

WORK (workspace/output) REAL array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

IWORK (workspace) INTEGER array, dimension (LIWORK)

LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N ).

INFO (output) INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Based on contributions by

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

Osni Marques, LBNL/NERSC, USA