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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgetrf (f07ar)

## Purpose

nag_lapack_zgetrf (f07ar) computes the $LU$ factorization of a complex $m$ by $n$ matrix.

## Syntax

[a, ipiv, info] = f07ar(a, 'm', m, 'n', n)
[a, ipiv, info] = nag_lapack_zgetrf(a, 'm', m, 'n', n)

## Description

nag_lapack_zgetrf (f07ar) forms the $LU$ factorization of a complex $m$ by $n$ matrix $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is lower triangular with unit diagonal elements (lower trapezoidal if $m>n$) and $U$ is upper triangular (upper trapezoidal if $m). Usually $A$ is square $\left(m=n\right)$, and both $L$ and $U$ are triangular. The function uses partial pivoting, with row interchanges.

## References

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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factors $L$ and $U$ from the factorization $A=PLU$; the unit diagonal elements of $L$ are not stored.
2:     $\mathrm{ipiv}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$int64int32nag_int array
The pivot indices that define the permutation matrix. At the $\mathit{i}$th step, if ${\mathbf{ipiv}}\left(\mathit{i}\right)>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row ${\mathbf{ipiv}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. ${\mathbf{ipiv}}\left(i\right)\le i$ indicates that, at the $i$th step, a row interchange was not required.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  ${\mathbf{info}}>0$
Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## Accuracy

The computed factors $L$ and $U$ are the exact factors of a perturbed matrix $A+E$, where
 $E ≤ c minm,n ε P L U ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{3}$ if $m=n$ (the usual case), $\frac{4}{3}{n}^{2}\left(3m-n\right)$ if $m>n$ and $\frac{4}{3}{m}^{2}\left(3n-m\right)$ if $m.
A call to this function with $m=n$ may be followed by calls to the functions:
• nag_lapack_zgetrs (f07as) to solve $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$;
• nag_lapack_zgecon (f07au) to estimate the condition number of $A$;
• nag_lapack_zgetri (f07aw) to compute the inverse of $A$.
The real analogue of this function is nag_lapack_dgetrf (f07ad).

## Example

This example computes the $LU$ factorization of the matrix $A$, where
 $A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i .$
```function f07ar_example

fprintf('f07ar example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];

[LU, ipiv, info] = f07ar(a);

disp('Details of factorization');
disp(LU);
disp('Pivot indices');
disp(double(ipiv'));

```
```f07ar example results

Details of factorization
-3.2900 - 2.3900i  -1.9100 + 4.4200i  -0.1400 - 1.3500i   1.7200 + 1.3500i
0.2376 + 0.2560i   4.8952 - 0.7114i  -0.4623 + 1.6966i   1.2269 + 0.6190i
-0.1020 - 0.7010i  -0.6691 + 0.3689i  -5.1414 - 1.1300i   0.9983 + 0.3850i
-0.5359 + 0.2707i  -0.2040 + 0.8601i   0.0082 + 0.1211i   0.1482 - 0.1252i

Pivot indices
3     2     3     4

```