NAG FL Interface
f01adf (real_​symm_​posdef_​inv_​noref)

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1 Purpose

f01adf calculates the approximate inverse of a real symmetric positive definite matrix, using a Cholesky factorization.

2 Specification

Fortran Interface
Subroutine f01adf ( n, a, lda, ifail)
Integer, Intent (In) :: n, lda
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*)
C Header Interface
#include <nag.h>
void  f01adf_ (const Integer *n, double a[], const Integer *lda, Integer *ifail)
The routine may be called by the names f01adf or nagf_matop_real_symm_posdef_inv_noref.

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=LLT, where L is lower triangular. It then computes L-1 and finally forms X as the product L-TL-1.

4 References

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

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the upper triangle of the n×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 a(i+1,j) for ij. The upper triangle of the original matrix is unchanged.
3: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01adf is called.
Constraint: ldan+1.
4: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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:
The matrix A is not positive definite, possibly due to rounding errors.
On entry, lda<n+1: lda, n=value, value.
On entry, n=value.
Constraint: n0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01adf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01adf 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 f01adf is approximately proportional to n3. f01adf calls routines f07fdf and f07fjf from LAPACK.

10 Example

This example finds the inverse of the 4×4 matrix:
( 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 ) .  

10.1 Program Text

Program Text (f01adfe.f90)

10.2 Program Data

Program Data (f01adfe.d)

10.3 Program Results

Program Results (f01adfe.r)