CGELS - solve overdetermined or underdetermined complex
linear systems involving an M-by-N matrix A, or its conju
gate-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 conju
gate-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 solu
tion of
an underdetermined system A * X = B.
3. If TRANS = 'C' and m >= n: find the minimum norm solu
tion of
an undetermined system A**H * X = B.
4. If TRANS = 'C' and m < n: find the least squares solu
tion 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.
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 over
written by the solution vectors, stored column
wise: 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 >= 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.
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
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