NAG Library Routine Document

F04ASF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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	$LDA < \max (1, N)$ .

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 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.

9 Example

This example solves the set of linear equations

A x = b