dgelss(3)

NAME

DGELSS - compute the minimum norm solution to a real lin

ear least squares problem

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 solu

tion 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 vec

tors, 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 sin

gular values which are greater than RCOND*S(1).

WORK (workspace/output) DOUBLE PRECISION array, dimen

sion (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 ille

gal value.
> 0: the algorithm for computing the SVD failed

to converge; if INFO = i, i off-diagonal elements of an interme

diate bidiagonal form did not converge to zero.

LAPACK version 3.0 15 June 2000

DGELSS(3)