# NAG Library Routine Document

## 1Purpose

f01abf calculates the accurate inverse of a real symmetric positive definite matrix, using a Cholesky factorization and iterative refinement.

## 2Specification

Fortran Interface
 Subroutine f01abf ( a, lda, n, b, ldb, z,
 Integer, Intent (In) :: lda, n, ldb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: a(lda,n), b(ldb,n) Real (Kind=nag_wp), Intent (Out) :: z(n)
#include nagmk26.h
 void f01abf_ ( double a[], const Integer *lda, const Integer *n, double b[], const Integer *ldb, double z[], Integer *ifail)

## 3Description

To compute the inverse $X$ of a real symmetric positive definite matrix $A$, f01abf first computes a Cholesky factorization of $A$ as $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular. An approximation to $X$ is found by computing ${L}^{-1}$ and then the product ${L}^{-\mathrm{T}}{L}^{-1}$. The residual matrix $R=I-AX$ is calculated using additional precision, and a correction $D$ to $X$ is found by solving $L{L}^{\mathrm{T}}D=R$. $X$ is replaced by $X+D$, and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.

## 4References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5Arguments

1:     $\mathbf{a}\left({\mathbf{lda}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
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.
2:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01abf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}+1$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{b}\left({\mathbf{ldb}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the lower triangle of the inverse matrix $X$, with ${x}_{ij}$ stored in ${\mathbf{b}}\left(i,j\right)$, for $i\ge j$.
5:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f01abf is called.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{n}}$.
6:     $\mathbf{z}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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.
${\mathbf{ifail}}=2$
The refinement process fails to converge, i.e., the matrix $A$ is ill-conditioned.
${\mathbf{ifail}}=3$
${\mathbf{n}}<1$, or ${\mathbf{lda}}<{\mathbf{n}}+1$, or ${\mathbf{ldb}}<{\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computed inverse should be correct to full machine accuracy. For a detailed error analysis see page 40 of Wilkinson and Reinsch (1971).

## 8Parallelism and Performance

f01abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01abf 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 time taken by f01abf is approximately proportional to ${n}^{3}$.

## 10Example

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

### 10.1Program Text

Program Text (f01abfe.f90)

### 10.2Program Data

Program Data (f01abfe.d)

### 10.3Program Results

Program Results (f01abfe.r)

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