f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zsytri (f07nwc)

## 1  Purpose

nag_zsytri (f07nwc) computes the inverse of a complex symmetric matrix $A$, where $A$ has been factorized by nag_zsytrf (f07nrc).

## 2  Specification

 #include #include
 void nag_zsytri (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, const Integer ipiv[], NagError *fail)

## 3  Description

nag_zsytri (f07nwc) is used to compute the inverse of a complex symmetric matrix $A$, the function must be preceded by a call to nag_zsytrf (f07nrc), which computes the Bunch–Kaufman factorization of $A$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for $X$.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$ for $X$.

## 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
On entry: details of the factorization of $A$, as returned by nag_zsytrf (f07nrc).
On exit: the factorization is overwritten by the $n$ by $n$ symmetric matrix ${A}^{-1}$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangle of ${A}^{-1}$ is stored in the upper triangular part of the array.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangle of ${A}^{-1}$ is stored in the lower triangular part of the array.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     ipiv[$\mathit{dim}$]const IntegerInput
Note: the dimension, dim, 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 nag_zsytrf (f07nrc).
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
$d\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$ 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 ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $\left|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{U}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|U\right|+\left|D\right|\left|{D}^{-1}\right|\right)$;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, $\left|D{L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{L}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|L\right|+\left|D\right|\left|{D}^{-1}\right|\right)$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

The total number of real floating point operations is approximately $\frac{8}{3}{n}^{3}$.
The real analogue of this function is nag_dsytri (f07mjc).

## 9  Example

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 nag_zsytrf (f07nrc).

### 9.1  Program Text

Program Text (f07nwce.c)

### 9.2  Program Data

Program Data (f07nwce.d)

### 9.3  Program Results

Program Results (f07nwce.r)