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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgetrf (f07ar)

## Purpose

nag_lapack_zgetrf (f07ar) computes the LU$LU$ factorization of a complex m$m$ by n$n$ 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$LU$ factorization of a complex m$m$ by n$n$ matrix A$A$ as A = PLU$A=PLU$, where P$P$ is a permutation matrix, L$L$ is lower triangular with unit diagonal elements (lower trapezoidal if m > n$m>n$) and U$U$ is upper triangular (upper trapezoidal if m < n$m). Usually A$A$ is square (m = n)$\left(m=n\right)$, and both L$L$ and U$U$ 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:     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$A$.

### 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${\mathbf{n}}\ge 0$.

lda

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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: m, 2: n, 3: a, 4: lda, 5: ipiv, 6: 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.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, U(i,i)$U\left(i,i\right)$ is exactly zero. The factorization has been completed, but the factor U$U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## Accuracy

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

The total number of real floating point operations is approximately (8/3)n3$\frac{8}{3}{n}^{3}$ if m = n$m=n$ (the usual case), (4/3)n2(3mn)$\frac{4}{3}{n}^{2}\left(3m-n\right)$ if m > n$m>n$ and (4/3)m2(3nm)$\frac{4}{3}{m}^{2}\left(3n-m\right)$ if m < n$m.
A call to this function with m = n$m=n$ may be followed by calls to the functions:
• nag_lapack_zgetrs (f07as) to solve AX = B$AX=B$, ATX = B${A}^{\mathrm{T}}X=B$ or AHX = B${A}^{\mathrm{H}}X=B$;
• nag_lapack_zgecon (f07au) to estimate the condition number of A$A$;
• nag_lapack_zgetri (f07aw) to compute the inverse of A$A$.
The real analogue of this function is nag_lapack_dgetrf (f07ad).

## Example

```function nag_lapack_zgetrf_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  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];
[aOut, ipiv, info] = nag_lapack_zgetrf(a)
```
```

aOut =

-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

ipiv =

3
2
3
4

info =

0

```
```function f07ar_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  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];
[aOut, ipiv, info] = f07ar(a)
```
```

aOut =

-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

ipiv =

3
2
3
4

info =

0

```