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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_corrmat_nearest_bounded (g02ab)

## Purpose

nag_correg_corrmat_nearest_bounded (g02ab) computes the nearest correlation matrix, in the Frobenius norm or weighted Frobenius norm, and optionally with bounds on the eigenvalues, to a given square, input matrix.

## Syntax

[g, w, x, iter, feval, nrmgrd, ifail] = g02ab(g, opt, alpha, w, 'n', n, 'errtol', errtol, 'maxits', maxits, 'maxit', maxit)
[g, w, x, iter, feval, nrmgrd, ifail] = nag_correg_corrmat_nearest_bounded(g, opt, alpha, w, 'n', n, 'errtol', errtol, 'maxits', maxits, 'maxit', maxit)

## Description

Finds the nearest correlation matrix $X$ by minimizing $\frac{1}{2}{‖G-X‖}^{2}$ where $G$ is an approximate correlation matrix.
The norm can either be the Frobenius norm or the weighted Frobenius norm $\frac{1}{2}{‖{W}^{\frac{1}{2}}\left(G-X\right){W}^{\frac{1}{2}}‖}_{F}^{2}$.
You can optionally specify a lower bound on the eigenvalues, $\alpha$, of the computed correlation matrix, forcing the matrix to be positive definite, $0<\alpha <1$.
Note that if the weights vary by several orders of magnitude from one another the algorithm may fail to converge.

## 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.
2:     $\mathrm{opt}$ – string (length ≥ 1)
Indicates the problem to be solved.
${\mathbf{opt}}=\text{'A'}$
The lower bound problem is solved.
${\mathbf{opt}}=\text{'W'}$
The weighted norm problem is solved.
${\mathbf{opt}}=\text{'B'}$
Both problems are solved.
Constraint: ${\mathbf{opt}}=\text{'A'}$, $\text{'W'}$ or $\text{'B'}$.
3:     $\mathrm{alpha}$ – double scalar
The value of $\alpha$.
If ${\mathbf{opt}}=\text{'W'}$, alpha need not be set.
Constraint: $0.0<{\mathbf{alpha}}<1.0$.
4:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
The square roots of the diagonal elements of $W$, that is the diagonal of ${W}^{\frac{1}{2}}$.
If ${\mathbf{opt}}=\text{'A'}$, w is not referenced and need not be set.
Constraint: ${\mathbf{w}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array w and 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 order 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: $0$
Specifies the maximum number of iterations to be used by the iterative scheme 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
2:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{opt}}=\text{'W'}$ or $\text{'B'}$, the array is scaled so $0<{\mathbf{w}}\left(\mathit{i}\right)\le 1$, for $\mathit{i}=1,2,\dots ,n$.
3:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{n}}\right)$ – double array
Contains the nearest correlation matrix.
4:     $\mathrm{iter}$int64int32nag_int scalar
The number of Newton steps taken.
5:     $\mathrm{feval}$int64int32nag_int scalar
The number of function evaluations of the dual problem.
6:     $\mathrm{nrmgrd}$ – double scalar
The norm of the gradient of the last Newton step.
7:     $\mathrm{ifail}$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: $0.0<{\mathbf{alpha}}<1.0$.
Constraint: $\mathit{ldg}\ge {\mathbf{n}}$.
Constraint: $\mathit{ldx}\ge {\mathbf{n}}$.
Constraint: ${\mathbf{n}}>0$.
On entry, all elements of w were not positive.
On entry, ${\mathbf{opt}}\ne \text{'A'}$, $\text{'W'}$ or $\text{'B'}$.
${\mathbf{ifail}}=2$
Newton iteration fails to converge in $_$ iterations. Increase maxit or check the call to the function.
W  ${\mathbf{ifail}}=3$
The machine precision is limiting convergence. In this instance the returned value of x may be useful.
W  ${\mathbf{ifail}}=4$
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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_bounded (g02ab). The total size of these arrays is $12×{\mathbf{n}}+3×{\mathbf{n}}×{\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}}×{\mathbf{n}}+6×{\mathbf{n}}+1,120+9×{\mathbf{n}}\right)$ double elements and $5×{\mathbf{n}}+3$ integer elements. All allocated memory is freed before return of nag_correg_corrmat_nearest_bounded (g02ab).

## Example

This example finds the nearest correlation matrix to:
 $G = 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2$
weighted by ${W}^{\frac{1}{2}}=\mathrm{diag}\left(100,20,20,20\right)$ with minimum eigenvalue $0.02$.
```function g02ab_example

fprintf('g02ab example results\n\n');

opt = 'b';
alpha = 0.02;
g = [2, -1,  0,  0;
-1,  2, -1,  0;
0, -1,  2, -1;
0,  0, -1,  2];
w = [100, 20, 20, 20];

% Calculate nearest correlation matrix
[g, w, x, iter, feval, nrmgrd, ifail] = ...
g02ab(g, opt, alpha, w);

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);
fprintf('\n\n Alpha: %30.3f\n', alpha);

[x, eig, info] = f08fa('n', 'u', x);
fprintf('\n Eigenvalues of x:\n');
disp(transpose(eig));

```
```g02ab example results

Nearest Correlation Matrix:
1.0000   -0.9187    0.0257    0.0086
-0.9187    1.0000   -0.3008    0.2270
0.0257   -0.3008    1.0000   -0.8859
0.0086    0.2270   -0.8859    1.0000

Number of Newton steps taken:   5
Number of function evaluations: 6

Alpha:                          0.020

Eigenvalues of x:
0.0392    0.1183    1.6515    2.1910

```