F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07PJF (DSPTRI)

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

F07PJF (DSPTRI) computes the inverse of a real symmetric indefinite matrix $A$, where $A$ has been factorized by F07PDF (DSPTRF), using packed storage.

## 2  Specification

 SUBROUTINE F07PJF ( UPLO, N, AP, IPIV, WORK, INFO)
 INTEGER N, IPIV(*), INFO REAL (KIND=nag_wp) AP(*), WORK(N) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dsptri.

## 3  Description

F07PJF (DSPTRI) is used to compute the inverse of a real symmetric indefinite matrix $A$, the routine must be preceded by a call to F07PDF (DSPTRF), which computes the Bunch–Kaufman factorization of $A$, using packed storage.
If ${\mathbf{UPLO}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$.
If ${\mathbf{UPLO}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$.

## 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.
${\mathbf{UPLO}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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 factorization of $A$ stored in packed form, as returned by F07PDF (DSPTRF).
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:     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: details of the interchanges and the block structure of $D$, as returned by F07PDF (DSPTRF).
5:     WORK(N) – REAL (KIND=nag_wp) arrayWorkspace
6:     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 ${\mathbf{INFO}}=i$, $d\left(i,i\right)$ is exactly zero; $D$ is singular and the inverse of $A$ cannot be computed.

## 7  Accuracy

The computed inverse $X$ satisfies a bound of the form
• if ${\mathbf{UPLO}}=\text{'U'}$, $\left|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{U}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|U\right|+\left|D\right|\left|{D}^{-1}\right|\right)$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|D{L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{L}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|L\right|+\left|D\right|\left|{D}^{-1}\right|\right)$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8  Further Comments

The total number of floating point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogues of this routine are F07PWF (ZHPTRI) for Hermitian matrices and F07QWF (ZSPTRI) for symmetric matrices.

## 9  Example

This example computes the inverse of the matrix $A$, where
 $A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .$
Here $A$ is symmetric indefinite, stored in packed form, and must first be factorized by F07PDF (DSPTRF).

### 9.1  Program Text

Program Text (f07pjfe.f90)

### 9.2  Program Data

Program Data (f07pjfe.d)

### 9.3  Program Results

Program Results (f07pjfe.r)