F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07FWF (ZPOTRI)

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

F07FWF (ZPOTRI) computes the inverse of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by F07FRF (ZPOTRF).

## 2  Specification

 SUBROUTINE F07FWF ( UPLO, N, A, LDA, INFO)
 INTEGER N, LDA, INFO COMPLEX (KIND=nag_wp) A(LDA,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zpotri.

## 3  Description

F07FWF (ZPOTRI) is used to compute the inverse of a complex Hermitian positive definite matrix $A$, the routine must be preceded by a call to F07FRF (ZPOTRF), which computes the Cholesky factorization of $A$.
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 ${L}^{-\mathrm{H}}\left({L}^{-1}\right)$.
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:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/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 F07FRF (ZPOTRF).
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:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07FWF (ZPOTRI) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     $\mathrm{INFO}$ – 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

F07FWF (ZPOTRI) is not threaded by NAG in any implementation.
F07FWF (ZPOTRI) 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 F07FJF (DPOTRI).

## 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 and must first be factorized by F07FRF (ZPOTRF).

### 10.1  Program Text

Program Text (f07fwfe.f90)

### 10.2  Program Data

Program Data (f07fwfe.d)

### 10.3  Program Results

Program Results (f07fwfe.r)