# NAG Library Routine Document

## 1Purpose

f07gjf (dpptri) computes the inverse of a real symmetric positive definite matrix $A$, where $A$ has been factorized by f07gdf (dpptrf), using packed storage.

## 2Specification

Fortran Interface
 Subroutine f07gjf ( uplo, n, ap, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ap(*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07gjf_ (const char *uplo, const Integer *n, double ap[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dpptri.

## 3Description

f07gjf (dpptri) is used to compute the inverse of a real symmetric positive definite matrix $A$, the routine must be preceded by a call to f07gdf (dpptrf), which computes the Cholesky factorization of $A$, using packed storage.
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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the Cholesky factor of $A$ stored in packed form, as returned by f07gdf (dpptrf).
On exit: the factorization is overwritten by the $n$ by $n$ matrix ${A}^{-1}$.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
4:     $\mathbf{info}$ – IntegerOutput
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

f07gjf (dpptri) 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 f07gwf (zpptri).

## 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, stored in packed form, and must first be factorized by f07gdf (dpptrf).

### 10.1Program Text

Program Text (f07gjfe.f90)

### 10.2Program Data

Program Data (f07gjfe.d)

### 10.3Program Results

Program Results (f07gjfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017