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 <nag.h>

void	f01abf_ (double a[], const Integer lda, const Integer n, double b[], const Integer ldb, double z[], Integer ifail)

The routine may be called by the names f01abf or nagf_matop_real_symm_posdef_inv.

3 Description

To compute the inverse

X

of a real symmetric positive definite matrix

A

, f01abf first computes a Cholesky factorization of

A

A = L L^{T}

, where

L

is lower triangular. An approximation to

X

is found by computing

L^{- 1}

and then the product

L^{- T} L^{- 1}

. The residual matrix

R = I - A X

is calculated using additional precision, and a correction

D

X

is found by solving

L L^{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.

4 References

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

5 Arguments

1: $a (lda, n)$ – Real (Kind=nag_wp) array Input/Output: On entry: the upper triangle of the $n \times n$ 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 $n + 1$ ; $x_{i j}$ is stored in $a (i + 1, j)$ for $i \geq j$ . The upper triangle of the original matrix is unchanged.
2: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f01abf is called.

Constraint: $lda \geq n + 1$ .
3: $n$ – Integer Input: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 1$ .
4: $b (ldb, n)$ – Real (Kind=nag_wp) array Output: On exit: the lower triangle of the inverse matrix $X$ , with $x_{i j}$ stored in $b (i, j)$ , for $i \geq j$ .
5: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f01abf is called.

Constraint: $ldb \geq n$ .
6: $z (n)$ – Real (Kind=nag_wp) array Workspace
7: $ifail$ – 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

- 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: $ldb \geq n$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

n^{3}

10 Example

This example finds the inverse of the

4 \times 4

matrix:

(\begin{array}{r} 5 & 7 & 6 & 5 \\ 7 & 10 & 8 & 7 \\ 6 & 8 & 10 & 9 \\ 5 & 7 & 9 & 10 \end{array}) .

f01ab: FL CL CPP AD

NAG FL Interfacef01abf (real_​symm_​posdef_​inv)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f01abf (real_symm_posdef_inv)