# NAG FL Interfacef07fjf (dpotri)

## 1Purpose

f07fjf computes the inverse of a real symmetric positive definite matrix $A$, where $A$ has been factorized by f07fdf.

## 2Specification

Fortran Interface
 Subroutine f07fjf ( uplo, n, a, lda, info)
 Integer, Intent (In) :: n, lda Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07fjf_ (const char *uplo, const Integer *n, double a[], const Integer *lda, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07fjf, nagf_lapacklin_dpotri or its LAPACK name dpotri.

## 3Description

f07fjf is used to compute the inverse of a real symmetric positive definite matrix $A$, the routine must be preceded by a call to f07fdf, which computes the Cholesky factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{T}}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.

## 4References

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

## 5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/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 upper triangular matrix $U$ if ${\mathbf{uplo}}=\text{'U'}$ or the lower triangular matrix $L$ if ${\mathbf{uplo}}=\text{'L'}$, as returned by f07fdf.
On exit: $U$ is overwritten by the upper triangle of ${A}^{-1}$ if ${\mathbf{uplo}}=\text{'U'}$; $L$ is overwritten by the lower triangle of ${A}^{-1}$ if ${\mathbf{uplo}}=\text{'L'}$.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07fjf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and 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.
${\mathbf{info}}>0$
Diagonal element $〈\mathit{\text{value}}〉$ of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of $A$ cannot be computed.

## 7Accuracy

The computed inverse $X$ satisfies
 $XA-I2≤cnεκ2A and AX-I2≤cnεκ2A ,$
where $c\left(n\right)$ is a modest function of $n$, $\epsilon$ is the machine precision and ${\kappa }_{2}\left(A\right)$ is the condition number of $A$ defined by
 $κ2A=A2A-12 .$

## 8Parallelism and Performance

f07fjf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogue of this routine is f07fwf.

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .$
Here $A$ is symmetric positive definite and must first be factorized by f07fdf.

### 10.1Program Text

Program Text (f07fjfe.f90)

### 10.2Program Data

Program Data (f07fjfe.d)

### 10.3Program Results

Program Results (f07fjfe.r)