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NAG Toolbox: nag_lapack_zgeqlf (f08cs)
Purpose
nag_lapack_zgeqlf (f08cs) computes a factorization of a complex by matrix .
Syntax
Description
nag_lapack_zgeqlf (f08cs) forms the factorization of an arbitrary rectangular complex 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
lower trapezoidal matrix and
is again an
by
unitary matrix. In the case where
the factorization can be expressed as
where
consists of the first
columns of
, and
the remaining
columns.
The matrix
is not formed explicitly but is represented as a product of
elementary reflectors (see
Representation of orthogonal or unitary matrices in the F08 Chapter Introduction for details). Functions are provided to work with
in this representation (see
Further Comments).
Note also that for any
, the information returned in the last
columns of the array
a represents a
factorization of the last
columns of the original matrix
.
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 lower triangle of the subarray
contains the
by
lower triangular matrix
.
If
, the elements on and below the
th superdiagonal contain the
by
lower 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 (see
Further Comments).
- 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 real floating-point operations is approximately if or if .
To form the unitary matrix
nag_lapack_zgeqlf (f08cs) may be followed by a call to
nag_lapack_zungql (f08ct):
[a, info] = f08ct(a, tau);
but note that the second dimension of the array
a must be at least
m, which may be larger than was required by
nag_lapack_zgeqlf (f08cs).
When
, it is often only the first
columns of
that are required, and they may be formed by the call:
[a, info] = f08ct(a, tau, 'k', n);
To apply
to an arbitrary complex rectangular matrix
,
nag_lapack_zgeqlf (f08cs) may be followed by a call to
nag_lapack_zunmql (f08cu). For example,
[c, info] = f08cu('Left','Conjugate Transpose', a, tau, c);
forms
, where
is
by
.
The real analogue of this function is
nag_lapack_dgeqlf (f08ce).
Example
This example solves the linear least squares problems
for
and
, where
is the
th column of the matrix
,
and
The solution is obtained by first obtaining a 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:
f08cs_example
function f08cs_example
fprintf('f08cs example results\n\n');
m = 6;
n = 4;
a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i, -0.05 + 0.41i;
-0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i, -0.81 + 0.56i;
0.62 - 0.46i, 1.01 + 0.02i, 0.63 - 0.17i, -1.11 + 0.60i;
-0.37 + 0.38i, 0.19 - 0.54i, -0.98 - 0.36i, 0.22 - 0.20i;
0.83 + 0.51i, 0.20 + 0.01i, -0.17 - 0.46i, 1.47 + 1.59i;
1.08 - 0.28i, 0.20 - 0.12i, -0.07 + 1.23i, 0.26 + 0.26i];
b = [-2.09 + 1.93i, 3.26 - 2.70i;
3.34 - 3.53i, -6.22 + 1.16i;
-4.94 - 2.04i, 7.94 - 3.13i;
0.17 + 4.23i, 1.04 - 4.26i;
-5.19 + 3.63i, -2.31 - 2.12i;
0.98 + 2.53i, -1.39 - 4.05i];
[ql, tau, info] = f08cs(a);
[c, info] = f08cu( ...
'Left', 'ConjTrans', ql, tau, b);
il = m-n+1;
[x, info] = f07ts( ...
'Lower', 'Notrans','Non-Unit', ql(il:m,:), c(il:m,:));
ncols = int64(80);
indent = int64(0);
[ifail] = x04db( ...
'General', ' ', x, 'Bracketed', 'F7.4', ...
'Least-squares solution(s)', 'Integer', 'Integer', ...
ncols, indent);
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(c(1:m-n,j));
end
fprintf('\nSquare root(s) of the residual sum(s) of squares\n');
fprintf('\t%11.2e %11.2e\n', rnorm(1), rnorm(2));
f08cs example results
Least-squares solution(s)
1 2
1 (-0.5044,-1.2179) ( 0.7629, 1.4529)
2 (-2.4281, 2.8574) ( 5.1570,-3.6089)
3 ( 1.4872,-2.1955) (-2.6518, 2.1203)
4 ( 0.4537, 2.6904) (-2.7606, 0.3318)
Square root(s) of the residual sum(s) of squares
6.88e-02 1.87e-01
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