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NAG Toolbox: nag_lapack_dorgqr (f08af)
Purpose
nag_lapack_dorgqr (f08af) generates all or part of the real orthogonal matrix
from a
factorization computed by
nag_lapack_dgeqrf (f08ae),
nag_lapack_dgeqpf (f08be) or
nag_lapack_dgeqp3 (f08bf).
Syntax
[
a,
info] = nag_lapack_dorgqr(
a,
tau, 'm',
m, 'n',
n, 'k',
k)
Description
nag_lapack_dorgqr (f08af) is intended to be used after a call to
nag_lapack_dgeqrf (f08ae),
nag_lapack_dgeqpf (f08be) or
nag_lapack_dgeqp3 (f08bf).
which perform a
factorization of a real matrix
. The orthogonal matrix
is represented as a product of elementary reflectors.
This function may be used to generate explicitly as a square matrix, or to form only its leading columns.
Usually
is determined from the
factorization of an
by
matrix
with
. The whole of
may be computed by:
[a, info] = f08af(a, tau, 'k', p);
(note that the array
a must have
columns) or its leading
columns by:
[a, info] = f08af(a(:,1:p), tau, 'k', p);
The columns of
returned by the last call form an orthonormal basis for the space spanned by the columns of
; thus
nag_lapack_dgeqrf (f08ae) followed by
nag_lapack_dorgqr (f08af) can be used to orthogonalize the columns of
.
The information returned by the
factorization functions also yields the
factorization of the leading
columns of
, where
. The orthogonal matrix arising from this factorization can be computed by:
[a, info] = f08af(a, tau, 'k', k);
or its leading
columns by:
[a, info] = f08af(a(:,1:p), tau, 'k', k);
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parameters
Compulsory Input Parameters
- 1:
– double array
-
The first dimension of the array
a must be at least
.
The second dimension of the array
a must be at least
.
Details of the vectors which define the elementary reflectors, as returned by
nag_lapack_dgeqrf (f08ae),
nag_lapack_dgeqpf (f08be) or
nag_lapack_dgeqp3 (f08bf).
- 2:
– double array
-
The dimension of the array
tau
must be at least
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the array
a.
, the order of the orthogonal matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
a.
, the number of columns of the matrix .
Constraint:
.
- 3:
– int64int32nag_int scalar
-
Default:
the dimension of the array
tau.
, the number of elementary reflectors whose product defines the matrix .
Constraint:
.
Output Parameters
- 1:
– double array
-
The first dimension of the array
a will be
.
The second dimension of the array
a will be
.
The by matrix .
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , parameter had an illegal value on entry. The parameters are numbered as follows:
1:
m, 2:
n, 3:
k, 4:
a, 5:
lda, 6:
tau, 7:
work, 8:
lwork, 9:
info.
It is possible that
info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
Accuracy
The computed matrix
differs from an exactly orthogonal matrix by a matrix
such that
where
is the
machine precision.
Further Comments
The total number of floating-point operations is approximately ; when , the number is approximately .
The complex analogue of this function is
nag_lapack_zungqr (f08at).
Example
This example forms the leading
columns of the orthogonal matrix
from the
factorization of the matrix
, where
The columns of
form an orthonormal basis for the space spanned by the columns of
.
Open in the MATLAB editor:
f08af_example
function f08af_example
fprintf('f08af example results\n\n');
a = [-0.57, -1.28, -0.39, 0.25;
-1.93, 1.08, -0.31, -2.14;
2.30, 0.24, 0.40, -0.35;
-1.93, 0.64, -0.66, 0.08;
0.15, 0.30, 0.15, -2.13;
-0.02, 1.03, -1.43, 0.50];
[a, tau, info] = f08ae(a);
[q, info] = f08af(a, tau);
disp('Orthogonal factor Q');
disp(q);
f08af example results
Orthogonal factor Q
-0.1576 0.6744 -0.4571 0.4489
-0.5335 -0.3861 0.2583 0.3898
0.6358 -0.2928 0.0165 0.1930
-0.5335 -0.1692 -0.0834 -0.2350
0.0415 -0.1593 0.1475 0.7436
-0.0055 -0.5064 -0.8339 0.0335
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