F01ABF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F01ABF

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.

+− Contents

    1  Purpose
    2  Specification
    3  Description
    4  References
    5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
+− 9  Example

1  Purpose

F01ABF calculates the accurate inverse of a real symmetric positive-definite matrix, using a Cholesky factorization and iterative refinement.

2  Specification

SUBROUTINE F01ABF (A, LDA, N, B, LDB, Z, IFAIL)
INTEGERLDA, N, LDB, IFAIL
double precisionA(LDA,N), B(LDB,N), Z(N)

3  Description

To compute the inverse X of a real symmetric positive-definite matrix A, F01ABF first computes a Cholesky factorization of A as A=LLT, where L is lower triangular. An approximation to X is found by computing L-1 and then the product L-TL-1. The residual matrix R=I-AX is calculated using additional precision, and a correction D to X is found by solving LLTD=R. X is replaced by X+D, and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.

4  References

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

5  Parameters

1:     A(LDA,N) – double precision 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; xij is stored in Ai+1j for i≥j. The upper triangle of the original matrix is unchanged.
2:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01ABF is called.
Constraint: LDA≥N+1.
3:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N≥1.
4:     B(LDB,N) – double precision arrayOutput
On exit: the lower triangle of the inverse matrix X, with xij stored in Bij, for i≥j.
5:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F01ABF is called.
Constraint: LDB≥N.
6:     Z(N) – double precision arrayWorkspace
7:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
On exit: IFAIL=0 unless the routine detects an error (see Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ 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 -1​ or ​1 is used it is essential to test the value of IFAIL on exit.

6  Error Indicators and Warnings

If on entry 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:
IFAIL=1
The matrix A is not positive-definite, possibly due to rounding errors.
IFAIL=2
The refinement process fails to converge, i.e., the matrix A is ill-conditioned.
IFAIL=3
N<1 , or LDA<N+1 , or LDB<N .

7  Accuracy

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

8  Further Comments

The time taken by F01ABF is approximately proportional to n3.

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 .

9.1  Program Text

Program Text (f01abfe.f)

9.2  Program Data

Program Data (f01abfe.d)

9.3  Program Results

Program Results (f01abfe.r)

F01ABF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009