# NAG Library Routine Document

## 1Purpose

f07nwf (zsytri) computes the inverse of a complex symmetric matrix $A$, where $A$ has been factorized by f07nrf (zsytrf).

## 2Specification

Fortran Interface
 Subroutine f07nwf ( uplo, n, a, lda, ipiv, work, info)
 Integer, Intent (In) :: n, lda, ipiv(*) Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Complex (Kind=nag_wp), Intent (Out) :: work(2*n) Character (1), Intent (In) :: uplo
#include <nagmk26.h>
 void f07nwf_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, const Integer ipiv[], Complex work[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zsytri.

## 3Description

f07nwf (zsytri) is used to compute the inverse of a complex symmetric matrix $A$, the routine must be preceded by a call to f07nrf (zsytrf), which computes the Bunch–Kaufman factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for $X$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$ for $X$.

## 4References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

## 5Arguments

1:     $\mathbf{uplo}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 f07nrf (zsytrf).
On exit: the factorization is overwritten by the $n$ by $n$ symmetric 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:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07nwf (zsytri) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{ipiv}\left(*\right)$ – 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 f07nrf (zsytrf).
6:     $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

$-999<{\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
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.

## 7Accuracy

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.

## 8Parallelism and Performance

f07nwf (zsytri) 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).

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i .$
Here $A$ is symmetric and must first be factorized by f07nrf (zsytrf).

### 10.1Program Text

Program Text (f07nwfe.f90)

### 10.2Program Data

Program Data (f07nwfe.d)

### 10.3Program Results

Program Results (f07nwfe.r)