Integer, Intent (In)	::	ldg, n, maxits, maxit, ldx
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	iter, feval
Real (Kind=nag_wp), Intent (In)	::	errtol
Real (Kind=nag_wp), Intent (Inout)	::	g(ldg,n), x(ldx,n)
Real (Kind=nag_wp), Intent (Out)	::	nrmgrd

C Header Interface

#include nagmk26.h

void	g02aaf_ ( double g[], const Integer ldg, const Integer n, const double errtol, const Integer maxits, const Integer maxit, double x[], const Integer ldx, Integer iter, Integer feval, double nrmgrd, Integer ifail)

3

Description

A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.

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

Arguments

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 $maxits \leq 0$ , $100$ 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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $ldg = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldg \geq n$ .

On entry, $ldx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldx \geq n$ .

On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .

$ifail = 2$: Newton iteration fails to converge in $〈value〉$ iterations.

$ifail = 3$: Machine precision is limiting convergence.
The array returned in x may still be of interest.

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

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

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

8

Parallelism and Performance

g02aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02aaf 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

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.

10

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 Document

g02aaf (corrmat_nearest)

▸▿ 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

1

Purpose

2

Specification