CGELS(l) ) CGELS(l)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
)
CHARACTER TRANS
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
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 follow‐
ing 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 vec‐
tors, 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 solu‐
tion vectors; if TRANS = 'C' and m >= n, rows 1 to M of B con‐
tain 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 col‐
umn.
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 >= 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 opti‐
mum 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 illegal value
LAPACK version 3.0 15 June 2000 CGELS(l)
