NAG FL Interface
f08zaf (dgglse)
1
Purpose
f08zaf solves a real linear equality-constrained least squares problem.
2
Specification
Fortran Interface
Subroutine f08zaf ( |
m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info) |
Integer, Intent (In) |
:: |
m, n, p, lda, ldb, lwork |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*), c(m), d(p) |
Real (Kind=nag_wp), Intent (Out) |
:: |
x(n), work(max(1,lwork)) |
|
C Header Interface
#include <nag.h>
void |
f08zaf_ (const Integer *m, const Integer *n, const Integer *p, double a[], const Integer *lda, double b[], const Integer *ldb, double c[], double d[], double x[], double work[], const Integer *lwork, Integer *info) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08zaf_ (const Integer &m, const Integer &n, const Integer &p, double a[], const Integer &lda, double b[], const Integer &ldb, double c[], double d[], double x[], double work[], const Integer &lwork, Integer &info) |
}
|
The routine may be called by the names f08zaf, nagf_lapackeig_dgglse or its LAPACK name dgglse.
3
Description
f08zaf solves the real linear equality-constrained least squares (LSE) problem
where
is an
by
matrix,
is a
by
matrix,
is an
element vector and
is a
element vector. It is assumed that
,
and
, where
. These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized
factorization of the matrices
and
.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
Eldèn L (1980) Perturbation theory for the least squares problem with linear equality constraints SIAM J. Numer. Anal. 17 338–350
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of columns of the matrices and .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the by matrix .
On exit:
a is overwritten.
-
5:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08zaf is called.
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the by matrix .
On exit:
b is overwritten.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08zaf is called.
Constraint:
.
-
8:
– Real (Kind=nag_wp) array
Input/Output
-
On entry: the right-hand side vector for the least squares part of the LSE problem.
On exit: the residual sum of squares for the solution vector is given by the sum of squares of elements ; the remaining elements are overwritten.
-
9:
– Real (Kind=nag_wp) array
Input/Output
-
On entry: the right-hand side vector for the equality constraints.
On exit:
d is overwritten.
-
10:
– Real (Kind=nag_wp) array
Output
-
On exit: the solution vector of the LSE problem.
-
11:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
12:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08zaf is called.
If
, 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.
Suggested value:
for optimal performance, , where is the optimal block size.
Constraint:
or .
-
13:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
The upper triangular factor associated with in the generalized factorization of the pair is singular, so that ; the least squares solution could not be computed.
-
The by part of the upper trapezoidal factor associated with in the generalized factorization of the pair is singular, so that the rank of the matrix () comprising the rows of and is less than ; the least squares solutions could not be computed.
7
Accuracy
For an error analysis, see
Anderson et al. (1992) and
Eldèn (1980). See also Section 4.6 of
Anderson et al. (1999).
8
Parallelism and Performance
f08zaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
When , the total number of floating-point operations is approximately ; if , the number reduces to approximately .
e04ncf/e04nca may also be used to solve LSE problems. It differs from
f08zaf in that it uses an iterative (rather than direct) method, and that it allows general upper and lower bounds to be specified for the variables
and the linear constraints
.
10
Example
This example solves the least squares problem
where
and
The constraints correspond to and .
10.1
Program Text
10.2
Program Data
10.3
Program Results