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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgels (f08an)

## Purpose

nag_lapack_zgels (f08an) solves linear least squares problems of the form
 $minx b-Ax2 or minx b-AHx2 ,$
where $A$ is an $m$ by $n$ complex matrix of full rank, using a $QR$ or $LQ$ factorization of $A$.

## Syntax

[a, b, info] = f08an(trans, a, b, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, info] = nag_lapack_zgels(trans, a, b, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

## Description

The following options are provided:
1. If ${\mathbf{trans}}=\text{'N'}$ and $m\ge n$: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
 $minx b-Ax2 .$
2. If ${\mathbf{trans}}=\text{'N'}$ and $m: find the minimum norm solution of an underdetermined system $Ax=b$.
3. If ${\mathbf{trans}}=\text{'C'}$ and $m\ge n$: find the minimum norm solution of an undetermined system ${A}^{\mathrm{H}}x=b$.
4. If ${\mathbf{trans}}=\text{'C'}$ and $m: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
 $minx b-AHx2 .$
Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call; they are stored as the columns of the $m$ by $r$ right-hand side matrix $B$ and the $n$ by $r$ solution matrix $X$.

## 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:     $\mathrm{trans}$ – string (length ≥ 1)
If ${\mathbf{trans}}=\text{'N'}$, the linear system involves $A$.
If ${\mathbf{trans}}=\text{'C'}$, the linear system involves ${A}^{\mathrm{H}}$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'C'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $A$.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The matrix $B$ of right-hand side vectors, stored in columns; b is $m$ by $r$ if ${\mathbf{trans}}=\text{'N'}$, or $n$ by $r$ if ${\mathbf{trans}}=\text{'C'}$.

### 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: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the array b.
$r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### 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 ${\mathbf{m}}\ge {\mathbf{n}}$, a stores details of its $QR$ factorization, as returned by nag_lapack_zgeqrf (f08as).
If ${\mathbf{m}}<{\mathbf{n}}$, a stores details of its $LQ$ factorization, as returned by nag_lapack_zgelqf (f08av).
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
b stores the solution vectors, $x$, stored in columns:
• if ${\mathbf{trans}}=\text{'N'}$ and $m\ge n$, or ${\mathbf{trans}}=\text{'C'}$ and $m, elements $1$ to $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)+1\right)$ to $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in that column;
• otherwise, elements $1$ to $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in each column of b contain the minimum norm solution vectors.
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: trans, 2: m, 3: n, 4: nrhs_p, 5: a, 6: lda, 7: b, 8: ldb, 9: work, 10: lwork, 11: 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.
${\mathbf{info}}>0$
If ${\mathbf{info}}=i$, diagonal element $i$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.

## Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

The total number of floating-point operations required to factorize $A$ is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ and $\frac{8}{3}{m}^{2}\left(3n-m\right)$ otherwise. Following the factorization the solution for a single vector $x$ requires $\mathit{O}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({m}^{2},{n}^{2}\right)\right)$ operations.
The real analogue of this function is nag_lapack_dgels (f08aa).

## Example

This example solves the linear least squares problem
 $minx b-Ax2 ,$
where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i$
and
 $b = -2.09+1.93i 3.34-3.53i -4.94-2.04i 0.17+4.23i -5.19+3.63i 0.98+2.53i .$
The square root of the residual sum of squares is also output.
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 f08an_example

fprintf('f08an example results\n\n');

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;
3.34 - 3.53i;
-4.94 - 2.04i;
0.17 + 4.23i;
-5.19 + 3.63i;
0.98 + 2.53i];
[m,n] = size(a);

% Solve the least squares problem min( norm2(b - Ax) ) for x
trans = 'No transpose';
[af, x, info] = f08an( ...
trans, a, b);

% Print Solution
fprintf('\nLeast Squares Solution:\n');
disp(transpose(x(1:n)));
fprintf('Square root of the residual sum of squares\n');
disp(norm(x(n+1:m),2));

```
```f08an example results

Least Squares Solution:
-0.5044 - 1.2179i  -2.4281 + 2.8574i   1.4872 - 2.1955i   0.4537 + 2.6904i

Square root of the residual sum of squares
0.0688

```