F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07AJF (DGETRI)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07AJF (DGETRI) computes the inverse of a real matrix $A$, where $A$ has been factorized by F07ADF (DGETRF).

## 2  Specification

 SUBROUTINE F07AJF ( N, A, LDA, IPIV, WORK, LWORK, INFO)
 INTEGER N, LDA, IPIV(*), LWORK, INFO REAL (KIND=nag_wp) A(LDA,*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name dgetri.

## 3  Description

F07AJF (DGETRI) is used to compute the inverse of a real matrix $A$, the routine must be preceded by a call to F07ADF (DGETRF), which computes the $LU$ factorization of $A$ as $A=PLU$. The inverse of $A$ is computed by forming ${U}^{-1}$ and then solving the equation $XPL={U}^{-1}$ for $X$.
Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $LU$ factorization of $A$, as returned by F07ADF (DGETRF).
On exit: the factorization is overwritten by the $n$ by $n$ matrix ${A}^{-1}$.
3:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07AJF (DGETRI) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
4:     IPIV($*$) – INTEGER arrayInput
Note: the dimension of the array IPIV must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the pivot indices, as returned by F07ADF (DGETRF).
5:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}=0$, ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimum performance.
6:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F07AJF (DGETRI) is called, unless ${\mathbf{LWORK}}=-1$, in which case a workspace query is assumed and the routine only calculates the optimal dimension of WORK (using the formula given below).
Suggested value: for optimum performance LWORK should be at least ${\mathbf{N}}×\mathit{nb}$, where $\mathit{nb}$ is the block size.
Constraint: ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ or ${\mathbf{LWORK}}=-1$.
7:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the $i$th diagonal element of the factor $U$ is zero, $U$ is singular, and the inverse of $A$ cannot be computed.

## 7  Accuracy

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

## 8  Further Comments

The total number of floating point operations is approximately $\frac{4}{3}{n}^{3}$.
The complex analogue of this routine is F07AWF (ZGETRI).

## 9  Example

This example computes the inverse of the matrix $A$, where
 $A= 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 .$
Here $A$ is nonsymmetric and must first be factorized by F07ADF (DGETRF).

### 9.1  Program Text

Program Text (f07ajfe.f90)

### 9.2  Program Data

Program Data (f07ajfe.d)

### 9.3  Program Results

Program Results (f07ajfe.r)