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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
 min ‖b − Ax‖2  or min ‖b − AHx‖2, x x
$minx ‖b-Ax‖2 or minx ‖b-AHx‖2 ,$
where A$A$ is an m$m$ by n$n$ complex matrix of full rank, using a QR$QR$ or LQ$LQ$ factorization of A$A$.

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

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

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrices B$B$ and X$X$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,m)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
If mn${\mathbf{m}}\ge {\mathbf{n}}$, a stores details of its QR$QR$ factorization, as returned by nag_lapack_zgeqrf (f08as).
If m < n${\mathbf{m}}<{\mathbf{n}}$, a stores details of its LQ$LQ$ factorization, as returned by nag_lapack_zgelqf (f08av).
2:     b(ldb, : $:$) – complex array
The first dimension of the array b will be max (1,m,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,m,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
b stores the solution vectors, x$x$, stored in columns:
• if trans = 'N'${\mathbf{trans}}=\text{'N'}$ and mn$m\ge n$, or trans = 'C'${\mathbf{trans}}=\text{'C'}$ and m < n$m, elements 1$1$ to min (m,n)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ 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 (min (m,n) + 1) $\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)+1\right)$ to max (m,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in that column;
• otherwise, elements 1$1$ to max (m,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in each column of b contain the minimum norm solution vectors.
3:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, diagonal element i$i$ of the triangular factor of A$A$ is zero, so that A$A$ 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$A$ is approximately (8/3) n2 (3mn) $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if mn$m\ge n$ and (8/3) m2 (3nm) $\frac{8}{3}{m}^{2}\left(3n-m\right)$ otherwise. Following the factorization the solution for a single vector x$x$ requires O(min (m2,n2)) $\mathit{O}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({m}^{2},{n}^{2}\right)\right)$ operations.
The real analogue of this function is nag_lapack_dgels (f08aa).

## Example

```function nag_lapack_zgels_example
trans = 'No transpose';
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
-0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
-0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i,  -1.11 + 0.6i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.2i;
0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.2 - 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];
% Solve the least squares problem min( norm2(b - Ax) ) for x
[a, b, info] = nag_lapack_zgels(trans, a, b);

% Print Solution
fprintf('\nLeast Squares Solution:\n');
disp(transpose(b(1:4)));
fprintf('Square root of the residual sum of squares\n');
disp(norm(b(5:6),2));
```
```

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

```
```function f08an_example
trans = 'No transpose';
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
-0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
-0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i,  -1.11 + 0.6i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.2i;
0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.2 - 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];
% Solve the least squares problem min( norm2(b - Ax) ) for x
[a, b, info] = f08an(trans, a, b);

% Print Solution
fprintf('\nLeast Squares Solution:\n');
disp(transpose(b(1:4)));
fprintf('Square root of the residual sum of squares\n');
disp(norm(b(5:6),2));
```
```

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

```