# zgels (l) - Linux Manuals

## NAME

ZGELS - 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

## SYNOPSIS

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

CHARACTER TRANS

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

COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )

## PURPOSE

ZGELS 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 = aqNaq 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 = aqNaq and m < n: find the minimum norm solution of
an underdetermined system A B.
3. If TRANS = aqCaq and m >= n: find the minimum norm solution of
an undetermined system A**H B.
4. If TRANS = aqCaq 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*1
= aqNaq: the linear system involves A;
= aqCaq: 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*16 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 ZGEQRF; if M < N, A is overwritten by details of its LQ factorization as returned by ZGELQF.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = aqNaq, or N-by-NRHS if TRANS = aqCaq. On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise: if TRANS = aqNaq 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 the modulus of elements N+1 to M in that column; if TRANS = aqNaq and m < n, rows 1 to N of B contain the minimum norm solution vectors; if TRANS = aqCaq and m >= n, rows 1 to M of B contain the minimum norm solution vectors; if TRANS = aqCaq 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 the modulus 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*16 array, dimension (MAX(1,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 illegal value
> 0: if INFO = i, the i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed.