SYNOPSIS

SUBROUTINE DGELSS(

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

    

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

    

DOUBLE PRECISION RCOND

    

DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

PURPOSE

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

ARGUMENTS

M (input) INTEGER

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

N (input) INTEGER

The number of columns of the matrix 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/output) DOUBLE PRECISION array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise.

LDA (input) INTEGER

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

B (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION

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) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

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

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= 1, and also: LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) 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.

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.