SGELS(1) LAPACK driver routine (version 3.2) SGELS(1)NAME
SGELS - solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its transpose, using a QR or LQ fac‐
torization 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 fac‐
torization 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 = 'T' and m >= n: find the minimum norm solution of
an undetermined system A**T * X = B.
4. If TRANS = 'T' and m < n: find the least squares solution 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 solution matrix X.
ARGUMENTS
TRANS (input) CHARACTER*1
= '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 over‐
written by details of its QR factorization as returned by SGE‐
QRF; if M < N, A is overwritten by details of its LQ factor‐
ization 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, if INFO = 0, 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 col‐
umn; 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 (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 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
> 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.
LAPACK driver routine (version 3November 2008 SGELS(1)
