NAG FL Interface
f08aaf (dgels)
1
Purpose
f08aaf solves linear least squares problems of the form
where
is an
by
real matrix of full rank, using a
or
factorization of
.
2
Specification
Fortran Interface
Subroutine f08aaf ( |
trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info) |
Integer, Intent (In) |
:: |
m, n, nrhs, lda, ldb, lwork |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
Character (1), Intent (In) |
:: |
trans |
|
C Header Interface
#include <nag.h>
void |
f08aaf_ (const char *trans, const Integer *m, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double b[], const Integer *ldb, double work[], const Integer *lwork, Integer *info, const Charlen length_trans) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08aaf_ (const char *trans, const Integer &m, const Integer &n, const Integer &nrhs, double a[], const Integer &lda, double b[], const Integer &ldb, double work[], const Integer &lwork, Integer &info, const Charlen length_trans) |
}
|
The routine may be called by the names f08aaf, nagf_lapackeig_dgels or its LAPACK name dgels.
3
Description
The following options are provided:
-
1.If and : find the least squares solution of an overdetermined system, i.e., solve the least squares problem
-
2.If and : find the minimum norm solution of an underdetermined system .
-
3.If and : find the minimum norm solution of an undetermined system .
-
4.If and : find the least squares solution of an overdetermined system, i.e., solve the least squares problem
Several right-hand side vectors and solution vectors can be handled in a single call; they are stored as the columns of the by right-hand side matrix and the by solution matrix .
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
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Character(1)
Input
-
On entry: if
, the linear system involves
.
If , the linear system involves .
Constraint:
or .
-
2:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
-
5:
– 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: if
,
a is overwritten by details of its
factorization, as returned by
f08aef.
If
,
a is overwritten by details of its
factorization, as returned by
f08ahf.
-
6:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08aaf is called.
Constraint:
.
-
7:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the matrix
of right-hand side vectors, stored in columns;
b is
by
if
, or
by
if
.
On exit:
b is overwritten by the solution vectors,
, stored in columns:
- if and , or and , elements to in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements to in that column;
- otherwise, elements to in each column of b contain the minimum norm solution vectors.
-
8:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08aaf is called.
Constraint:
.
-
9:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
10:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08aaf 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 .
-
11:
– 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.
-
Diagonal element of the triangular factor of is zero, so that does not have full rank; the least squares solution could not be computed.
7
Accuracy
See Section 4.5 of
Anderson et al. (1999) for details of error bounds.
8
Parallelism and Performance
f08aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aaf 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.
The total number of floating-point operations required to factorize is approximately if and otherwise. Following the factorization the solution for a single vector requires operations.
The complex analogue of this routine is
f08anf.
10
Example
This example solves the linear least squares problem
where
The square root of the residual sum of squares is also output.
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
10.1
Program Text
10.2
Program Data
10.3
Program Results