# NAG Library Function Document

## 1Purpose

nag_dsytri (f07mjc) computes the inverse of a real symmetric indefinite matrix $A$, where $A$ has been factorized by nag_dsytrf (f07mdc).

## 2Specification

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

## 3Description

nag_dsytri (f07mjc) is used to compute the inverse of a real symmetric indefinite matrix $A$, the function must be preceded by a call to nag_dsytrf (f07mdc), 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 $X$.

## 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:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/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_dsytrf (f07mdc).
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:    $\mathbf{pda}$IntegerInput
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:    $\mathbf{ipiv}\left[\mathit{dim}\right]$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_dsytrf (f07mdc).
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
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 ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, $\left|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{U}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|U\right|+\left|D\right|\left|{D}^{-1}\right|\right)$;
• 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.

## 8Parallelism and Performance

nag_dsytri (f07mjc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogues of this function are nag_zhetri (f07mwc) for Hermitian matrices and nag_zsytri (f07nwc) for symmetric matrices.

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .$
Here $A$ is symmetric indefinite and must first be factorized by nag_dsytrf (f07mdc).

### 10.1Program Text

Program Text (f07mjce.c)

### 10.2Program Data

Program Data (f07mjce.d)

### 10.3Program Results

Program Results (f07mjce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017