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NAG Toolbox: nag_correg_corrmat_nearest_bounded (g02ab)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

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 12G-X2 where G is an approximate correlation matrix.
The norm can either be the Frobenius norm or the weighted Frobenius norm 12 W12 G-X W12 F 2 .
You can optionally specify a lower bound on the eigenvalues, α, of the computed correlation matrix, forcing the matrix to be positive definite, 0<α<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:     gldgn – double array
ldg, the first dimension of the array, must satisfy the constraint ldgn.
G, the initial matrix.
2:     opt – string (length ≥ 1)
Indicates the problem to be solved.
opt='A'
The lower bound problem is solved.
opt='W'
The weighted norm problem is solved.
opt='B'
Both problems are solved.
Constraint: opt='A', 'W' or 'B'.
3:     alpha – double scalar
The value of α.
If opt='W', alpha need not be set.
Constraint: 0.0<alpha<1.0.
4:     wn – double array
The square roots of the diagonal elements of W, that is the diagonal of W12.
If opt='A', w is not referenced and need not be set.
Constraint: wi>0.0, for i=1,2,,n.

Optional Input Parameters

1:     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: n>0.
2:     errtol – double scalar
Default: 0.0 
The termination tolerance for the Newton iteration. If errtol0.0 then n×machine precision is used.
3:     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 maxits0, 2×n is used.
4:     maxit int64int32nag_int scalar
Default: 0 
Specifies the maximum number of Newton iterations.
If maxit0, 200 is used.

Output Parameters

1:     gldgn – double array
2:     wn – double array
If opt='W' or 'B', the array is scaled so 0<wi1, for i=1,2,,n.
3:     xldxn – double array
Contains the nearest correlation matrix.
4:     iter int64int32nag_int scalar
The number of Newton steps taken.
5:     feval int64int32nag_int scalar
The number of function evaluations of the dual problem.
6:     nrmgrd – double scalar
The norm of the gradient of the last Newton step.
7:     ifail int64int32nag_int scalar
ifail=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.

   ifail=1
Constraint: 0.0<alpha<1.0.
Constraint: ldgn.
Constraint: ldxn.
Constraint: n>0.
On entry, all elements of w were not positive.
On entry, opt'A', 'W' or 'B'.
   ifail=2
Newton iteration fails to converge in _ iterations. Increase maxit or check the call to the function.
W  ifail=3
The machine precision is limiting convergence. In this instance the returned value of x may be useful.
W  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.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

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

Further Comments

Arrays are internally allocated by nag_correg_corrmat_nearest_bounded (g02ab). The total size of these arrays is 12×n+3×n×n+max2×n×n+6×n+1,120+9×n double elements and 5×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 W12 = diag 100 , 20 , 20 , 20  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


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