NAME

cgels - solve overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A

SYNOPSIS

SUBROUTINE CGELS(
TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO )

void cgels(char trans, long int m, long int n, long int nrhs,
complex ∗ca, long int lda, complex ∗cb, long int ldb, long int ∗info)

CHARACTER TRANS

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

COMPLEX A( LDA, ∗ ), B( LDB, ∗ ), WORK( LWORK )

PURPOSE

CGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-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 solution 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 solution of
   an underdetermined system A ∗ X = B.
 
3. If TRANS = ’C’ and m >= n:  find the minimum norm solution of
   an undetermined system A∗∗H ∗ X = B.
 
4. If TRANS = ’C’ and m < n:  find the least squares solution of
   an overdetermined system, i.e., solve the least squares problem
                minimize || B - A∗∗H ∗ 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 solution matrix X.
 

ARGUMENTS

TRANS   (input) CHARACTER
= ’N’: the linear system involves A;
= ’C’: the linear system involves A∗∗H.

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) COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A. if M >= N, A is overwritten by details of its QR factorization as returned by CGEQRF; if M <  N, A is overwritten by details of its LQ factorization as returned by CGELQF.

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

B       (input/output) COMPLEX 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 = ’C’. 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 = ’C’ and m >= n, rows 1 to M of B contain the minimum norm solution vectors; if TRANS = ’C’ 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) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK   (input) INTEGER
The dimension of the array WORK. LWORK >= min(M,N) + MAX(1,M,N,NRHS). For optimal performance, LWORK >= min(M,N) + MAX(1,M,N,NRHS) ∗ NB where NB is the optimum block size.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value

Sun, Inc.  —  Last change: 20 Sep 1996