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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgeqlf (f08cs)

Purpose

nag_lapack_zgeqlf (f08cs) computes a $QL$ factorization of a complex $m$ by $n$ matrix $A$.

Syntax

[a, tau, info] = f08cs(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_zgeqlf(a, 'm', m, 'n', n)

Description

nag_lapack_zgeqlf (f08cs) forms the $QL$ factorization of an arbitrary rectangular complex $m$ by $n$ matrix.
If $m\ge n$, the factorization is given by:
 $A = Q 0 L ,$
where $L$ is an $n$ by $n$ lower triangular matrix and $Q$ is an $m$ by $m$ unitary matrix. If $m the factorization is given by
 $A = QL ,$
where $L$ is an $m$ by $n$ lower trapezoidal matrix and $Q$ is again an $m$ by $m$ unitary matrix. In the case where $m>n$ the factorization can be expressed as
 $A = Q1 Q2 0 L = Q2 L ,$
where ${Q}_{1}$ consists of the first $m-n$ columns of $Q$, and ${Q}_{2}$ the remaining $n$ columns.
The matrix $Q$ is not formed explicitly but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).
Note also that for any $k, the information returned in the last $k$ columns of the array a represents a $QL$ factorization of the last $\mathrm{k}$ columns of the original matrix $A$.

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $A$.

Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If $m\ge n$, the lower triangle of the subarray ${\mathbf{a}}\left(m-n+1:m,1:n\right)$ contains the $n$ by $n$ lower triangular matrix $L$.
If $m\le n$, the elements on and below the $\left(n-m\right)$th superdiagonal contain the $m$ by $n$ lower trapezoidal matrix $L$. The remaining elements, with the array tau, represent the unitary matrix $Q$ as a product of elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     $\mathrm{tau}\left(:\right)$ – complex array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
The scalar factors of the elementary reflectors (see Further Comments).
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ 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 $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{8}{3}{m}^{2}\left(3n-m\right)$ if $m.
To form the unitary matrix $Q$ 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 $m\ge n$, it is often only the first $n$ columns of $Q$ that are required, and they may be formed by the call:
```[a, info] = f08ct(a, tau, 'k', n);
```
To apply $Q$ to an arbitrary complex rectangular matrix $C$, 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 $C={Q}^{\mathrm{H}}C$, where $C$ is $m$ by $p$.
The real analogue of this function is nag_lapack_dgeqlf (f08ce).

Example

This example solves the linear least squares problems
 $minx bj - Axj 2 , ​ j=1,2$
for ${x}_{1}$ and ${x}_{2}$, where ${b}_{j}$ is the $j$th column of the matrix $B$,
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i$
and
 $B = -2.09+1.93i 3.26-2.70i 3.34-3.53i -6.22+1.16i -4.94-2.04i 7.94-3.13i 0.17+4.23i 1.04-4.26i -5.19+3.63i -2.31-2.12i 0.98+2.53i -1.39-4.05i .$
The solution is obtained by first obtaining a $QL$ factorization of the matrix $A$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```function f08cs_example

fprintf('f08cs example results\n\n');

% Find least squares solution of Ax=B (m>n) via QL factorization.
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];

% Compute the QL factorization of A
[ql, tau, info] = f08cs(a);

% LX = Q^H B = C; compute C = (Q^H)*B
[c, info] = f08cu( ...
'Left', 'ConjTrans', ql, tau, b);

% Least-squares solution X = L^-1 C (lower n part)
il = m-n+1;
[x, info] = f07ts( ...
'Lower', 'Notrans','Non-Unit', ql(il:m,:), c(il:m,:));

% Print least-squares solutions
ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
'General', ' ', x, 'Bracketed', 'F7.4', ...
'Least-squares solution(s)', 'Integer', 'Integer', ...
ncols, indent);

% Compute estimates of the square roots of the residual sums of squares.
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
```