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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgeqlf (f08ce)

## Purpose

nag_lapack_dgeqlf (f08ce) computes a $QL$ factorization of a real $m$ by $n$ matrix $A$.

## Syntax

[a, tau, info] = f08ce(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgeqlf(a, 'm', m, 'n', n)

## Description

nag_lapack_dgeqlf (f08ce) forms the $QL$ factorization of an arbitrary rectangular real $m$ 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 $m$ orthogonal 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 $m$ orthogonal 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 orthogonal matrix $Q$ as a product of elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     $\mathrm{tau}\left(:\right)$ – double 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 floating-point operations is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m.
To form the orthogonal matrix $Q$ nag_lapack_dgeqlf (f08ce) may be followed by a call to nag_lapack_dorgql (f08cf):
```[a, info] = f08cf(a(:,1:m), 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_dgeqlf (f08ce).
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] = f08cf(a, tau);
```
To apply $Q$ to an arbitrary real rectangular matrix $C$, nag_lapack_dgeqlf (f08ce) may be followed by a call to nag_lapack_dormql (f08cg). For example,
```[c, info] = f08cg('Left','Transpose', a(:,1:min(m,n)), tau, c);
```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $m$ by $p$.
The complex analogue of this function is nag_lapack_zgeqlf (f08cs).

## 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 $B$,
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 and B= -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 .$
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 f08ce_example

fprintf('f08ce example results\n\n');

% Find least squares solution of Ax=B (m>n) via QL factorization.
m = 6;
n = 4;
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,  0.3,   0.15, -2.13;
-0.02,  1.03, -1.43,  0.5];
b = [-2.67,  0.41;
-0.55, -3.10;
3.34, -4.01;
-0.77,  2.76;
0.48, -6.17;
4.10,  0.21];

% Compute the QL factorization of A
[ql, tau, info] = f08ce(a);

% LX = Q^T B = C; compute C = (Q^T)*B
[c, info] = f08cg( ...
'Left', 'Transpose', ql, tau, b);

% Least-squares solution X = L^-1 C (lower n part)
il = m-n+1;
[x, info] = f07te( ...
'Lower', 'NoTrans', 'Non-Unit', ql(il:m,:), c(il:m,:));

% Print least-squares solutions
[ifail] = x04ca( ...
'General', ' ', x, 'Least-squares solution(s)');
% Compute estimates of the square roots of the residual sums of squares
rnorm = zeros(2,1);
for j=1:2
rnorm(j) = norm(c(1:2,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));

```
```f08ce example results

Least-squares solution(s)
1          2
1      1.5339    -1.5753
2      1.8707     0.5559
3     -1.5241     1.3119
4      0.0392     2.9585

Square root(s) of the residual sum(s) of squares
2.22e-02       1.38e-02
```