NAG Library Routine Document

G02AAF

G02AAF applies an inexact Newton method to a dual formulation of the problem, as described by Qi and Sun (2006). It applies the improvements suggested by Borsdorf and Higham (2010).

4 References

Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis 30(1) 94–107

Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl 29(2) 360–385

5 Parameters

1: G(LDG,N) – REAL (KIND=nag_wp) arrayInput/Output: On entry: $G$ , the initial matrix.

On exit: a symmetric matrix $\frac{1}{2} (G + G^{T})$ with the diagonal set to $I$ .
2: LDG – INTEGERInput: On entry: the first dimension of the array G as declared in the (sub)program from which G02AAF is called.
Constraint: $LDG \geq N$ .
3: N – INTEGERInput: On entry: the size of the matrix $G$ .
Constraint: $N > 0$ .
4: ERRTOL – REAL (KIND=nag_wp)Input: On entry: the termination tolerance for the Newton iteration. If $ERRTOL \leq 0.0$ then $N \times \sqrt{machine precision}$ is used.
5: MAXITS – INTEGERInput: On entry: MAXITS specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If $MAXITS \leq 0$ , $2 \times N$ is used.
6: MAXIT – INTEGERInput: On entry: specifies the maximum number of Newton iterations.
If $MAXIT \leq 0$ , $200$ is used.
7: X(LDX,N) – REAL (KIND=nag_wp) arrayOutput: On exit: contains the nearest correlation matrix.
8: LDX – INTEGERInput: On entry: the first dimension of the array X as declared in the (sub)program from which G02AAF is called.
Constraint: $LDX \geq N$ .
9: ITER – INTEGEROutput: On exit: the number of Newton steps taken.
10: FEVAL – INTEGEROutput: On exit: the number of function evaluations of the dual problem.
11: NRMGRD – REAL (KIND=nag_wp)Output: On exit: the norm of the gradient of the last Newton step.
12: 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:

$IFAIL = 1$

On entry,	$N \leq 0$ ,
or	$LDG < N$ ,
or	$LDX < N$ .

$IFAIL = 2$: The routine fails to converge in MAXIT iterations. Increase MAXIT or check the call to the routine.

$IFAIL = 3$: Machine precision is limiting convergence. In this instance the returned value of X may be useful.

$IFAIL = 4$: An internal eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

The returned accuracy is controlled by ERRTOL and limited by machine precision.

8 Further Comments

Arrays are internally allocated by G02AAF. The total size of these arrays is

11 \times N + 3 \times N \times N + \max (2 \times N \times N + 6 \times N + 1, 120 + 9 \times N)

real elements and

5 \times N + 3

integer elements.

9 Example

This example finds the nearest correlation matrix to:

G = (\begin{matrix} 2 & - 1 & 0 & 0 \\ - 1 & 2 & - 1 & 0 \\ 0 & - 1 & 2 & - 1 \\ 0 & 0 & - 1 & 2 \end{matrix})

NAG Library Routine DocumentG02AAF

+− Contents

1 Purpose

2 Specification

3 Description