sgels(3)

NAME

SGELS - solve overdetermined or underdetermined real lin

ear systems involving an M-by-N matrix A, or its transpose, using

a QR or LQ factorization of A

SYNOPSIS

SUBROUTINE SGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK,
LWORK, INFO )
    CHARACTER     TRANS
    INTEGER       INFO, LDA, LDB, LWORK, M, N, NRHS
    REAL          A( LDA, * ), B( LDB, * ), WORK( * )

PURPOSE

SGELS solves overdetermined or underdetermined real linear

systems involving an M-by-N matrix A, or its transpose, using a

QR or LQ factorization of A. It is assumed that A has full rank.

The following options are provided:

1. If TRANS = 'N' and m >= n: find the least squares so

lution of
an overdetermined system, i.e., solve the least squares

problem
minimize || B - A*X ||.

2. If TRANS = 'N' and m < n: find the minimum norm solu

tion of
an underdetermined system A * X = B.

3. If TRANS = 'T' and m >= n: find the minimum norm solu

tion of
an undetermined system A**T * X = B.

4. If TRANS = 'T' and m < n: find the least squares solu

tion of
an overdetermined system, i.e., solve the least squares

problem
minimize || B - A**T * X ||.

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.

ARGUMENTS

TRANS (input) CHARACTER

= 'N': the linear system involves A;
= 'T': the linear system involves A**T.

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) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if M >=

N, A is overwritten by details of its QR factorization as re

turned by SGEQRF; if M < N, A is overwritten by details of its

LQ factorization as returned by SGELQF.

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 matrix B of right hand side vectors,

stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if

TRANS = 'T'. On exit, B is overwritten by the solution vectors,

stored columnwise: if TRANS = 'N' and m >= n, rows 1 to n of B

contain the least squares solution vectors; the residual sum of

squares for the solution in each column is given by the sum of

squares of elements N+1 to M in that column; if TRANS = 'N' and m

< n, rows 1 to N of B contain the minimum norm solution vectors;

if TRANS = 'T' and m >= n, rows 1 to M of B contain the minimum

norm solution vectors; if TRANS = 'T' and m < n, rows 1 to M of B

contain the least squares solution vectors; the residual sum of

squares for the solution in each column is given by the sum of

squares of elements M+1 to N in that column.

LDB (input) INTEGER
The leading dimension of the array B. LDB >=

MAX(1,M,N).

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 >= max( 1,

MN + max( MN, NRHS ) ). For optimal performance, LWORK >= max(

1, MN + max( MN, NRHS )*NB ). where MN = min(M,N) and NB is the

optimum block size.

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

LAPACK version 3.0 15 June 2000

SGELS(3)