# NAG Library Routine Document

## 1Purpose

f07wjf (dpftri) computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by f07wdf (dpftrf) stored in Rectangular Full Packed (RFP) format.

## 2Specification

Fortran Interface
 Subroutine f07wjf ( uplo, n, ar, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ar(n*(n+1)/2) Character (1), Intent (In) :: transr, uplo
#include nagmk26.h
 void f07wjf_ ( const char *transr, const char *uplo, const Integer *n, double ar[], Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by its LAPACK name dpftri.

## 3Description

f07wjf (dpftri) is used to compute the inverse of a real symmetric positive definite matrix $A$, stored in RFP format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction. The routine must be preceded by a call to f07wdf (dpftrf), which computes the Cholesky factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{T}}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.
Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## 5Arguments

1:     $\mathbf{transr}$ – Character(1)Input
On entry: specifies whether the RFP representation of $A$ is normal or transposed.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'T'}$
The matrix $A$ is stored in transposed RFP format.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the Cholesky factorization of $A$ stored in RFP format, as returned by f07wdf (dpftrf).
On exit: the factorization is overwritten by the $n$ by $n$ matrix ${A}^{-1}$ stored in RFP format.
5:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.
${\mathbf{info}}>0$
The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. There is no routine specifically designed to invert a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling f07mjf (dsytri).

## 7Accuracy

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
 $κ2A=A2A-12 .$

## 8Parallelism and Performance

f07wjf (dpftri) 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 floating-point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogue of this routine is f07wwf (zpftri).

## 10Example

This example computes the inverse of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .$
Here $A$ is symmetric positive definite, stored in RFP format, and must first be factorized by f07wdf (dpftrf).

### 10.1Program Text

Program Text (f07wjfe.f90)

### 10.2Program Data

Program Data (f07wjfe.d)

### 10.3Program Results

Program Results (f07wjfe.r)

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