NAG Library Routine Document

f01abf (real_symm_posdef_inv)

1
Purpose

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

2
Specification

Fortran Interface
Subroutine f01abf ( a, lda, n, b, ldb, z, ifail)
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)
C Header Interface
#include <nagmk26.h>
void  f01abf_ (double a[], const Integer *lda, const Integer *n, double b[], const Integer *ldb, double z[], Integer *ifail)

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
Arguments

1:     aldan – 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; xij is stored in ai+1j for ij. 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: ldan+1.
3:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n1.
4:     bldbn – Real (Kind=nag_wp) arrayOutput
On exit: the lower triangle of the inverse matrix X, with xij stored in bij, for ij.
5:     ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f01abf is called.
Constraint: ldbn.
6:     zn – Real (Kind=nag_wp) arrayWorkspace
7:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 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 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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

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 failed to converge. The matrix A is ill-conditioned.
ifail=3
On entry, lda=value and n=value.
Constraint: lda>n.
On entry, ldb=value and n=value.
Constraint: ldbn.
On entry, n=value.
Constraint: n1.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
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.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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
Parallelism 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.

9
Further Comments

The time taken by f01abf is approximately proportional to n3.

10
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 .  

10.1
Program Text

Program Text (f01abfe.f90)

10.2
Program Data

Program Data (f01abfe.d)

10.3
Program Results

Program Results (f01abfe.r)