SGELSY - compute the minimum-norm solution to a real lin
ear 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 condition 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 anni
hilated 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
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 submatrix R11 in the QR factor
ization with pivoting of A, whose estimated condi
tion 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 complete orthogonal fac
torization of A.
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
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 ille
gal 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
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