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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgetri (f07aw)

## Purpose

nag_lapack_zgetri (f07aw) computes the inverse of a complex matrix A$A$, where A$A$ has been factorized by nag_lapack_zgetrf (f07ar).

## Syntax

[a, info] = f07aw(a, ipiv, 'n', n)
[a, info] = nag_lapack_zgetri(a, ipiv, 'n', n)

## Description

nag_lapack_zgetri (f07aw) is used to compute the inverse of a complex matrix A$A$, the function must be preceded by a call to nag_lapack_zgetrf (f07ar), which computes the LU$LU$ factorization of A$A$ as A = PLU$A=PLU$. The inverse of A$A$ is computed by forming U1${U}^{-1}$ and then solving the equation XPL = U1$XPL={U}^{-1}$ for X$X$.

## References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

## Parameters

### Compulsory Input Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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 LU$LU$ factorization of A$A$, as returned by nag_lapack_zgetrf (f07ar).
2:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The pivot indices, as returned by nag_lapack_zgetrf (f07ar).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the arrays a, ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factorization stores the n$n$ by n$n$ matrix A1${A}^{-1}$.
2:     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: n, 2: a, 3: lda, 4: ipiv, 5: work, 6: lwork, 7: 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${\mathbf{info}}=i$, the i$i$th diagonal element of the factor U$U$ is zero, U$U$ is singular, and the inverse of A$A$ cannot be computed.

## Accuracy

The computed inverse X$X$ satisfies a bound of the form:
 |XA − I| ≤ c(n)ε|X|P|L||U| , $|XA-I|≤c(n)ε|X|P|L||U| ,$
where c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision.
Note that a similar bound for |AXI|$|AX-I|$ cannot be guaranteed, although it is almost always satisfied. See Du Croz and Higham (1992).

The total number of real floating point operations is approximately (16/3)n3$\frac{16}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dgetri (f07aj).

## Example

function nag_lapack_zgetri_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];
% Factorize a
[a, ipiv, info] = nag_lapack_zgetrf(a);

% Compute inverse of a
[a, info] = nag_lapack_zgetri(a, ipiv)

a =

0.0757 - 0.4324i   1.6512 - 3.1342i   1.2663 + 0.0418i   3.8181 + 1.1195i
-0.1942 + 0.0798i  -1.1900 - 0.1426i  -0.2401 - 0.5889i  -0.0101 - 1.4969i
-0.0957 - 0.0491i   0.7371 - 0.4290i   0.3224 + 0.0776i   0.6887 + 0.7891i
0.3702 - 0.5040i   3.7253 - 3.1813i   1.7014 + 0.7267i   3.9367 + 3.3255i

info =

0

function f07aw_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];
% Factorize a
[a, ipiv, info] = f07ar(a);

% Compute inverse of a
[a, info] = f07aw(a, ipiv)

a =

0.0757 - 0.4324i   1.6512 - 3.1342i   1.2663 + 0.0418i   3.8181 + 1.1195i
-0.1942 + 0.0798i  -1.1900 - 0.1426i  -0.2401 - 0.5889i  -0.0101 - 1.4969i
-0.0957 - 0.0491i   0.7371 - 0.4290i   0.3224 + 0.0776i   0.6887 + 0.7891i
0.3702 - 0.5040i   3.7253 - 3.1813i   1.7014 + 0.7267i   3.9367 + 3.3255i

info =

0