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NAG Toolbox: nag_lapack_zgerqf (f08cv)
Purpose
nag_lapack_zgerqf (f08cv) computes an RQ factorization of a complex by matrix .
Syntax
Description
nag_lapack_zgerqf (f08cv) forms the
factorization of an arbitrary rectangular real
by
matrix. If
, the factorization is given by
where
is an
by
lower triangular matrix and
is an
by
unitary matrix. If
the factorization is given by
where
is an
by
upper trapezoidal matrix and
is again an
by
unitary matrix. In the case where
the factorization can be expressed as
where
consists of the first
rows of
and
the remaining
rows.
The matrix
is not formed explicitly, but is represented as a product of
elementary reflectors (see the
F08 Chapter Introduction for details). Functions are provided to work with
in this representation (see
Further Comments).
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
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parameters
Compulsory Input Parameters
- 1:
– complex array
-
The first dimension of the array
a must be at least
.
The second dimension of the array
a must be at least
.
The by matrix .
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the array
a.
, the number of rows of the matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
a.
, the number of columns of the matrix .
Constraint:
.
Output Parameters
- 1:
– complex array
-
The first dimension of the array
a will be
.
The second dimension of the array
a will be
.
If
, the upper triangle of the subarray
contains the
by
upper triangular matrix
.
If
, the elements on and above the
th subdiagonal contain the
by
upper trapezoidal matrix
; the remaining elements, with the array
tau, represent the unitary matrix
as a product of
elementary reflectors (see
Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
- 2:
– complex array
-
The dimension of the array
tau will be
The scalar factors of the elementary reflectors.
- 3:
– 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:
a, 4:
lda, 5:
tau, 6:
work, 7:
lwork, 8:
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 factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
Further Comments
The total number of floating-point operations is approximately if , or if .
To form the unitary matrix
nag_lapack_zgerqf (f08cv) may be followed by a call to
nag_lapack_zungrq (f08cw):
[a, info] = f08cw(a, tau, 'k', min(m,n));
but note that the first dimension of the array
a must be at least
n, which may be larger than was required by
nag_lapack_zgerqf (f08cv). When
, it is often only the first
rows of
that are required and they may be formed by the call:
[a, info] = f08cw(a, tau);
To apply
to an arbitrary real rectangular matrix
,
nag_lapack_zgerqf (f08cv) may be followed by a call to
nag_lapack_zunmrq (f08cx). For example:
[a, c, info] = f08cx('Left','C', a, tau, c);
forms
, where
is
by
.
The real analogue of this function is
nag_lapack_dgerqf (f08ch).
Example
This example finds the minimum norm solution to the underdetermined equations
where
and
The solution is obtained by first obtaining an factorization of the matrix .
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.
Open in the MATLAB editor:
f08cv_example
function f08cv_example
fprintf('f08cv example results\n\n');
m = 3;
n = 4;
a = [ 0.28 - 0.36i, 0.50 - 0.86i, -0.77 - 0.48i, 1.58 + 0.66i;
-0.50 - 1.10i, -1.21 + 0.76i, -0.32 - 0.24i, -0.27 - 1.15i;
0.36 - 0.51i, -0.07 + 1.33i, -0.75 + 0.47i, -0.08 + 1.01i];
b = [ -1.35 + 0.19i;
9.41 - 3.56i;
-7.57 + 6.93i];
[rq, tau, info] = f08cv(a);
c = zeros(n, 1);
il = n - m + 1;
[c(il:n,:), info] = f07ts( ...
'Upper', 'No transpose','Non-Unit', rq(:,il:n), b);
[rq, x, info] = f08cx( ...
'Left', 'Conjugate Transpose', rq, tau, c);
fprintf('Minimum-norm solution\n');
disp(x);
f08cv example results
Minimum-norm solution
-2.8501 + 6.4683i
1.6264 - 0.7799i
6.9290 + 4.6481i
1.4048 + 3.2400i
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