F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07GJF (DPPTRI)

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

F07GJF (DPPTRI) computes the inverse of a real symmetric positive definite matrix $A$, where $A$ has been factorized by F07GDF (DPPTRF), using packed storage.

## 2  Specification

 SUBROUTINE F07GJF ( UPLO, N, AP, INFO)
 INTEGER N, INFO REAL (KIND=nag_wp) AP(*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpptri.

## 3  Description

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 ${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.

## 4  References

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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – 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:     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 Cholesky factor is zero; the Cholesky factor is singular and the inverse of $A$ cannot be computed.

## 7  Accuracy

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
 $κ2A=A2A-12 .$

## 8  Further Comments

The total number of floating point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogue of this routine is F07GWF (ZPPTRI).

## 9  Example

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).

### 9.1  Program Text

Program Text (f07gjfe.f90)

### 9.2  Program Data

Program Data (f07gjfe.d)

### 9.3  Program Results

Program Results (f07gjfe.r)