
SGELS(l) ) SGELS(l)
NAME
SGELS  solve overdetermined or underdetermined real linear systems involving an MbyN
matrix A, or its transpose, using a QR or LQ factorization 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 MbyN
matrix A, or its 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 = '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 MbyNRHS right hand side matrix B and the NbyNRHS
solution matrix X.
ARGUMENTS
TRANS (input) CHARACTER
= '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 MbyN matrix A. On exit, if M >= N, A is overwritten by details of
its QR factorization as returned by SGEQRF; if M < N, A is overwritten by details
of its LQ factorization 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 Mby
NRHS if TRANS = 'N', or NbyNRHS if TRANS = 'T'. 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 = '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 (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 ith argument had an illegal value
LAPACK version 3.0 15 June 2000 SGELS(l) 
