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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_corrmat_nearest (g02aa)

## Purpose

nag_correg_corrmat_nearest (g02aa) computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.

## Syntax

[g, x, iter, feval, nrmgrd, ifail] = g02aa(g, 'n', n, 'errtol', errtol, 'maxits', maxits, 'maxit', maxit)
[g, x, iter, feval, nrmgrd, ifail] = nag_correg_corrmat_nearest(g, 'n', n, 'errtol', errtol, 'maxits', maxits, 'maxit', maxit)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 23: errtol, maxits and maxit were made optional

## Description

A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
nag_correg_corrmat_nearest (g02aa) 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).

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{g}\left(\mathit{ldg},{\mathbf{n}}\right)$ – double array
ldg, the first dimension of the array, must satisfy the constraint $\mathit{ldg}\ge {\mathbf{n}}$.
$G$, the initial matrix.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array g and the second dimension of the array g. (An error is raised if these dimensions are not equal.)
The size of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
2:     $\mathrm{errtol}$ – double scalar
Default: $0.0$
The termination tolerance for the Newton iteration. If ${\mathbf{errtol}}\le 0.0$ then  is used.
3:     $\mathrm{maxits}$int64int32nag_int scalar
Default: $0$
maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If ${\mathbf{maxits}}\le 0$, $2×{\mathbf{n}}$ is used.
4:     $\mathrm{maxit}$int64int32nag_int scalar
Default: $0$
Specifies the maximum number of Newton iterations.
If ${\mathbf{maxit}}\le 0$, $200$ is used.

### Output Parameters

1:     $\mathrm{g}\left(\mathit{ldg},{\mathbf{n}}\right)$ – double array
A symmetric matrix $\frac{1}{2}\left(G+{G}^{\mathrm{T}}\right)$ with the diagonal set to $I$.
2:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{n}}\right)$ – double array
Contains the nearest correlation matrix.
3:     $\mathrm{iter}$int64int32nag_int scalar
The number of Newton steps taken.
4:     $\mathrm{feval}$int64int32nag_int scalar
The number of function evaluations of the dual problem.
5:     $\mathrm{nrmgrd}$ – double scalar
The norm of the gradient of the last Newton step.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
Constraint: $\mathit{ldg}\ge {\mathbf{n}}$.
Constraint: $\mathit{ldx}\ge {\mathbf{n}}$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
Newton iteration fails to converge in $_$ iterations.
W  ${\mathbf{ifail}}=3$
Machine precision is limiting convergence.
The array returned in x may still be of interest.
${\mathbf{ifail}}=4$
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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

Arrays are internally allocated by nag_correg_corrmat_nearest (g02aa). The total size of these arrays is $11×{\mathbf{n}}+3×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ real elements and $5×{\mathbf{n}}+3$ integer elements.

## Example

This example finds the nearest correlation matrix to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$
```function g02aa_example

fprintf('g02aa example results\n\n');

g = [ 2, -1,  0,  0;
-1,  2, -1,  0;
0, -1,  2, -1;
0,  0, -1,  2];

[g, x, iter, feval, nrmgrd, ifail] = ...
g02aa(g);

fprintf('\n     Nearest Correlation Matrix\n');
disp(x);
fprintf('\n Number of Newton steps taken:   %d\n', iter);
fprintf(' Number of function evaluations: %d\n', feval);
if (nrmgrd > 4*x02aj)
fprintf(' Norm of gradient of last Newton step: %6.4f\n', nrmgrd);
end

```
```g02aa example results

Nearest Correlation Matrix
1.0000   -0.8084    0.1916    0.1068
-0.8084    1.0000   -0.6562    0.1916
0.1916   -0.6562    1.0000   -0.8084
0.1068    0.1916   -0.8084    1.0000

Number of Newton steps taken:   3
Number of function evaluations: 4
Norm of gradient of last Newton step: 0.0000
```