F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

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

F01ADF calculates the approximate inverse of a real symmetric positive definite matrix, using a Cholesky factorization.

## 2  Specification

 SUBROUTINE F01ADF ( N, A, LDA, IFAIL)
 INTEGER N, LDA, IFAIL REAL (KIND=nag_wp) A(LDA,*)

## 3  Description

To compute the inverse $X$ of a real symmetric positive definite matrix $A$, F01ADF first computes a Cholesky factorization of $A$ as $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular. It then computes ${L}^{-1}$ and finally forms $X$ as the product ${L}^{-\mathrm{T}}{L}^{-1}$.
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     A(LDA,$*$) – REAL (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: the upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$. The elements of the array below the diagonal need not be set.
On exit: the lower triangle of the inverse matrix $X$ is stored in the elements of the array below the diagonal, in rows $2$ to $n+1$; ${x}_{ij}$ is stored in ${\mathbf{A}}\left(i+1,j\right)$ for $i\ge j$. The upper triangle of the original matrix is unchanged.
3:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01ADF is called.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}+1$.
4:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
The matrix $A$ is not positive definite, possibly due to rounding errors.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{N}}<0$, or ${\mathbf{LDA}}<{\mathbf{N}}+1$.

## 7  Accuracy

The accuracy of the computed inverse depends on the conditioning of the original matrix. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

The time taken by F01ADF is approximately proportional to ${n}^{3}$. F01ADF calls routines F07FDF (DPOTRF) and F07FJF (DPOTRI) from LAPACK.

## 9  Example

This example finds the inverse of the $4$ by $4$ matrix:
 $5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 .$