SGELSY(1) LAPACK driver routine (version 3.2) SGELSY(1)NAME
SGELSY - computes the minimum-norm solution to a real linear least
squares problem
SYNOPSIS
SUBROUTINE SGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
LWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
REAL RCOND
INTEGER JPVT( * )
REAL A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
SGELSY computes the minimum-norm solution to a real linear least
squares problem:
minimize || A * X - B ||
using a complete orthogonal factorization of A. A is an M-by-N matrix
which may be rank-deficient.
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.
The routine first computes a QR factorization with column pivoting:
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated con‐
dition number is less than 1/RCOND. The order of R11, RANK, is the
effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated by
orthogonal transformations from the right, arriving at the complete
orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first RANK columns of Q.
This routine is basically identical to the original xGELSX except three
differences:
o The call to the subroutine xGEQPF has been substituted by the
the call to the subroutine xGEQP3. This subroutine is a Blas-3
version of the QR factorization with column pivoting.
o Matrix B (the right hand side) is updated with Blas-3.
o The permutation of matrix B (the right hand side) is faster and
more simple.
ARGUMENTS
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
matrices B and X. NRHS >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A has been overwritten
by details of its complete orthogonal factorization.
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 M-by-NRHS right hand side matrix B. On exit, the
N-by-NRHS solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).
JPVT (input/output) INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
to the front of AP, otherwise column i is a free column. On
exit, if JPVT(i) = k, then the i-th column of AP was the k-th
column of A.
RCOND (input) REAL
RCOND is used to determine the effective rank of A, which is
defined as the order of the largest leading triangular subma‐
trix R11 in the QR factorization with pivoting of A, whose
estimated condition number < 1/RCOND.
RANK (output) INTEGER
The effective rank of A, i.e., the order of the submatrix R11.
This is the same as the order of the submatrix T11 in the com‐
plete orthogonal factorization of A.
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. The unblocked strategy
requires that: LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), where MN =
min( M, N ). The block algorithm requires that: LWORK >= MAX(
MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper bound on
the blocksize returned by ILAENV for the routines SGEQP3,
STZRZF, STZRQF, SORMQR, and SORMRZ. 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.
FURTHER DETAILS
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
LAPACK driver routine (version 3November 2008 SGELSY(1)
