NAG Library Routine Document

F04ASF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

F04ASF calculates the accurate solution of a set of real symmetric positive definite linear equations with a single right-hand side,

A x = b

, using a Cholesky factorization and iterative refinement.

2 Specification

SUBROUTINE F04ASF (

A, LDA, B, N, C, WK1, WK2, IFAIL)

INTEGER	LDA, N, IFAIL
REAL (KIND=nag_wp)	A(LDA,*), B(max(1,N)), C(max(1,N)), WK1(max(1,N)), WK2(max(1,N))

3 Description

Given a set of real linear equations

A x = b

, where

A

is a symmetric positive definite matrix, F04ASF first computes a Cholesky factorization of

A

A = L L^{T}

where

L

is lower triangular. An approximation to

x

is found by forward and backward substitution. The residual vector

r = b - A x

is then calculated using additional precision and a correction

d

x

is found by solving

L L^{T} d = r

x

is then replaced by

x + d

, and this iterative refinement of the solution is repeated until machine accuracy is obtained.

4 References

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

5 Parameters

1: $A (LDA *)$ – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .

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 elements of the array below the diagonal are overwritten; the upper triangle of $A$ is unchanged.
2: $LDA$ – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F04ASF is called.

Constraint: $LDA \geq \max (1, N)$ .
3: $B (\max (1, N))$ – REAL (KIND=nag_wp) arrayInput: Note: the dimension of the array B must be at least $\max (1, N)$ .

On entry: the right-hand side vector $b$ .
4: $N$ – INTEGERInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $N \geq 0$ .
5: $C (\max (1, N))$ – REAL (KIND=nag_wp) arrayOutput: On exit: the solution vector $x$ .
6: $WK1 (\max (1, N))$ – REAL (KIND=nag_wp) arrayWorkspace
7: $WK2 (\max (1, N))$ – REAL (KIND=nag_wp) arrayWorkspace
8: $IFAIL$ – 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.
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.

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.

$IFAIL = 2$: Iterative refinement fails to improve the solution, i.e., the matrix $A$ is too ill-conditioned.

$IFAIL = 3$

On entry,	$N < 0$ ,
or	$LDA < \max (1, N)$ .

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

8 Parallelism and Performance

F04ASF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F04ASF 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 F04ASF is approximately proportional to

n^{3}

The routine must not be called with the same name for parameters B and C.

10 Example

This example solves the set of linear equations

A x = b