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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_nearest_correlation_h_weight (g02aj)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_nearest_correlation_h_weight (g02aj) computes the nearest correlation matrix, using element-wise weighting in the Frobenius norm and optionally with bounds on the eigenvalues, to a given square, input matrix.


[g, h, x, iter, norm_p, ifail] = g02aj(g, alpha, h, 'n', n, 'errtol', errtol, 'maxit', maxit)
[g, h, x, iter, norm_p, ifail] = nag_nearest_correlation_h_weight(g, alpha, h, 'n', n, 'errtol', errtol, 'maxit', maxit)


nag_nearest_correlation_h_weight (g02aj) finds the nearest correlation matrix, X, to an approximate correlation matrix, G, using element-wise weighting, this minimizes H G-X F , where C=AB denotes the matrix C with elements Cij=Aij×Bij.
You can optionally specify a lower bound on the eigenvalues, α, of the computed correlation matrix, forcing the matrix to be strictly positive definite, if 0<α<1.
Zero elements in H should be used when you wish to put no emphasis on the corresponding element of G. The algorithm scales H so that the maximum element is 1. It is this scaled matrix that is used in computing the norm above and for the stopping criteria described in Accuracy.
Note that if the elements in H vary by several orders of magnitude from one another the algorithm may fail to converge.


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
Jiang K, Sun D and Toh K-C (To appear) An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP
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


Compulsory Input Parameters

1:     gldgn – double array
ldg, the first dimension of the array, must satisfy the constraint ldgn.
G, the initial matrix.
2:     alpha – double scalar
The value of α.
If alpha<0.0, 0.0 is used.
Constraint: alpha<1.0.
3:     hldhn – double array
ldh, the first dimension of the array, must satisfy the constraint ldhn.
The matrix of weights H.
Constraint: hij0.0, for all i and j=1,2,,n, ij.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays h, g and the second dimension of the arrays h, 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 iteration. If errtol0.0 then n×machine precision is used. See Accuracy for further details.
3:     maxit int64int32nag_int scalar
Default: 0 
Specifies the maximum number of iterations to be used.
If maxit0, 200 is used.

Output Parameters

1:     gldgn – double array
2:     hldhn – double array
A symmetric matrix 12 H+HT with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is 1.0.
3:     xldxn – double array
Contains the nearest correlation matrix.
4:     iter int64int32nag_int scalar
The number of iterations taken.
5:     norm_p – double scalar
The value of HG-XF after the final iteration.
6:     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.

Constraint: n>0.
Constraint: ldgn.
Constraint: ldhn.
Constraint: ldxn.
Constraint: alpha<1.0.
On entry, one or more of the off-diagonal elements of H were negative.
Function fails to converge in _ iterations.
Increase maxit or check the call to the function.
W  ifail=8
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG with details of your call.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The returned accuracy is controlled by errtol and limited by machine precision. If ei is the value of norm_p at the ith iteration, that is
ei = HG-XF ,  
where H has been scaled as described above, then the algorithm terminates when:
ei-ei-1 1+ maxei,ei-1 errtol .  

Further Comments

Arrays are internally allocated by nag_nearest_correlation_h_weight (g02aj). The total size of these arrays is 15×n+5×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_nearest_correlation_h_weight (g02aj).


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:
H = 0.0 10.0 0.0 0.0 10.0 0.0 1.5 1.5 0.0 1.5 0.0 0.0 0.0 1.5 0.0 0.0  
with minimum eigenvalue 0.04.
function g02aj_example

fprintf('g02aj example results\n\n');

g = [ 2, -1,  0,  0;
     -1,  2, -1,  0;
      0, -1,  2, -1;
      0,  0, -1,  2];
h = [ 0.0, 10.0,  0.0,  0.0;
     10.0,  0.0,  1.5,  1.5;
      0.0,  1.5,  0.0,  0.0;
      0.0,  1.5,  0.0,  0.0];
alpha = 0.04;
[g, h, x, iter, norm_p, ifail] = ...
  g02aj(g, alpha, h);

fprintf('\nReturned H Matrix\n');
fprintf('Nearest Correlation Matrix\n');
fprintf('Number of iterations taken:     %d\n', iter);
fprintf('Norm value: %26.4f\n', norm_p);
fprintf('Alpha:      %26.4f\n', alpha);

[~, w, info] = f08fa('n', 'u', x);

fprintf('\nEigenvalues of X\n');

g02aj example results

Returned H Matrix
         0    1.0000         0         0
    1.0000         0    0.1500    0.1500
         0    0.1500         0         0
         0    0.1500         0         0

Nearest Correlation Matrix
    1.0000   -0.9229    0.7734    0.0026
   -0.9229    1.0000   -0.7843   -0.0000
    0.7734   -0.7843    1.0000   -0.0615
    0.0026   -0.0000   -0.0615    1.0000

Number of iterations taken:     66
Norm value:                     0.1183
Alpha:                          0.0400

Eigenvalues of X
    0.0769    0.2637    1.0031    2.6563

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