F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07GWF (ZPPTRI)

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

F07GWF (ZPPTRI) computes the inverse of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by F07GRF (ZPPTRF), using packed storage.

## 2  Specification

 SUBROUTINE F07GWF ( UPLO, N, AP, INFO)
 INTEGER N, INFO COMPLEX (KIND=nag_wp) AP(*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zpptri.

## 3  Description

F07GWF (ZPPTRI) is used to compute the inverse of a complex Hermitian positive definite matrix $A$, the routine must be preceded by a call to F07GRF (ZPPTRF), which computes the Cholesky factorization of $A$, using packed storage.
If ${\mathbf{UPLO}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{H}}$.
If ${\mathbf{UPLO}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{H}}\left({L}^{-1}\right)$.

## 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:     $\mathrm{UPLO}$ – CHARACTER(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{UPLO}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     $\mathrm{AP}\left(*\right)$ – COMPLEX (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 F07GRF (ZPPTRF).
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:     $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.

## 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  Parallelism and Performance

F07GWF (ZPPTRI) is not threaded by NAG in any implementation.
F07GWF (ZPPTRI) 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 real floating-point operations is approximately $\frac{8}{3}{n}^{3}$.
The real analogue of this routine is F07GJF (DPPTRI).

## 10  Example

This example computes the inverse of the matrix $A$, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
Here $A$ is Hermitian positive definite, stored in packed form, and must first be factorized by F07GRF (ZPPTRF).

### 10.1  Program Text

Program Text (f07gwfe.f90)

### 10.2  Program Data

Program Data (f07gwfe.d)

### 10.3  Program Results

Program Results (f07gwfe.r)