F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07MWF (ZHETRI)

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

F07MWF (ZHETRI) computes the inverse of a complex Hermitian indefinite matrix $A$, where $A$ has been factorized by F07MRF (ZHETRF).

## 2  Specification

 SUBROUTINE F07MWF ( UPLO, N, A, LDA, IPIV, WORK, INFO)
 INTEGER N, LDA, IPIV(*), INFO COMPLEX (KIND=nag_wp) A(LDA,*), WORK(N) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zhetri.

## 3  Description

F07MWF (ZHETRI) is used to compute the inverse of a complex Hermitian indefinite matrix $A$, the routine must be preceded by a call to F07MRF (ZHETRF), which computes the Bunch–Kaufman factorization of $A$.
If ${\mathbf{UPLO}}=\text{'U'}$, $A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{H}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for $X$.
If ${\mathbf{UPLO}}=\text{'L'}$, $A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{H}}{P}^{\mathrm{T}}XPL={D}^{-1}$ for $X$.

## 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.
${\mathbf{UPLO}}=\text{'U'}$
$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, 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: details of the factorization of $A$, as returned by F07MRF (ZHETRF).
On exit: the factorization is overwritten by the $n$ by $n$ Hermitian matrix ${A}^{-1}$.
If ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of ${A}^{-1}$ is stored in the upper triangular part of the array.
If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of ${A}^{-1}$ is stored in the lower triangular part of the array.
4:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07MWF (ZHETRI) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     $\mathrm{IPIV}\left(*\right)$ – 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 F07MRF (ZHETRF).
6:     $\mathrm{WORK}\left({\mathbf{N}}\right)$ – COMPLEX (KIND=nag_wp) arrayWorkspace
7:     $\mathrm{INFO}$ – 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.
${\mathbf{INFO}}>0$
Element $〈\mathit{\text{value}}〉$ of the diagonal 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 ${\mathbf{UPLO}}=\text{'U'}$, $\left|D{U}^{\mathrm{H}}{P}^{\mathrm{T}}XPU-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{U}^{\mathrm{H}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|U\right|+\left|D\right|\left|{D}^{-1}\right|\right)$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|D{L}^{\mathrm{H}}{P}^{\mathrm{T}}XPL-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{L}^{\mathrm{H}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|L\right|+\left|D\right|\left|{D}^{-1}\right|\right)$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

F07MWF (ZHETRI) is not threaded by NAG in any implementation.
F07MWF (ZHETRI) 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 F07MJF (DSYTRI).

## 10  Example

This example computes the inverse of the matrix $A$, where
 $A= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i .$
Here $A$ is Hermitian indefinite and must first be factorized by F07MRF (ZHETRF).

### 10.1  Program Text

Program Text (f07mwfe.f90)

### 10.2  Program Data

Program Data (f07mwfe.d)

### 10.3  Program Results

Program Results (f07mwfe.r)