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NAG Toolbox: nag_nearest_correlation_h_weight (g02aj)

Purpose

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.

Syntax

[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)

Description

nag_nearest_correlation_h_weight (g02aj) finds the nearest correlation matrix, XX, to an approximate correlation matrix, GG, using element-wise weighting, this minimizes H(GX)F H (G-X) F , where C = ABC=AB denotes the matrix CC with elements Cij = Aij × BijCij=Aij×Bij.
You can optionally specify a lower bound on the eigenvalues, αα, of the computed correlation matrix, forcing the matrix to be strictly positive definite, if 0 < α < 10<α<1.
Zero elements in HH should be used when you wish to put no emphasis on the corresponding element of GG. The algorithm scales HH so that the maximum element is 11. It is this scaled matrix that is used in computing the norm above and for the stopping criteria described in Section [Accuracy].
Note that if the elements in HH 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
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

Parameters

Compulsory Input Parameters

1:     g(ldg,n) – double array
ldg, the first dimension of the array, must satisfy the constraint ldgnldgn.
GG, the initial matrix.
2:     alpha – double scalar
The value of αα.
If alpha < 0.0alpha<0.0, 0.00.0 is used.
Constraint: alpha < 1.0alpha<1.0.
3:     h(ldh,n) – double array
ldh, the first dimension of the array, must satisfy the constraint ldhnldhn.
The matrix of weights HH.
Constraint: h(i,j)0.0hij0.0, for all ii and j = 1,2,,nj=1,2,,n, ijij.

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 GG.
Constraint: n > 0n>0.
2:     errtol – double scalar
The termination tolerance for the iteration. If errtol0.0errtol0.0 then n × sqrt(machine precision)n×machine precision is used. See Section [Accuracy] for further details.
Default: 0.00.0 
3:     maxit – int64int32nag_int scalar
Specifies the maximum number of iterations to be used.
If maxit0maxit0, 200200 is used.
Default: 00 

Input Parameters Omitted from the MATLAB Interface

ldg ldh ldx

Output Parameters

1:     g(ldg,n) – double array
ldgnldgn.
A symmetric matrix (1/2)(G + GT)12(G+GT) with the diagonal set to II.
2:     h(ldh,n) – double array
ldhnldhn.
A symmetric matrix (1/2) (H + HT) 12 (H+HT) with its diagonal elements set to zero and the remaining elements scaled so that the maximum element is 1.01.0.
3:     x(ldx,n) – double array
ldxnldxn.
Contains the nearest correlation matrix.
4:     iter – int64int32nag_int scalar
The number of iterations taken.
5:     norm_p – double scalar
The value of H(GX)FH(G-X)F after the final iteration.
6:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1
Constraint: n > 0n>0.
  ifail = 2ifail=2
Constraint: ldgnldgn.
  ifail = 3ifail=3
Constraint: ldhnldhn.
  ifail = 4ifail=4
Constraint: ldxnldxn.
  ifail = 5ifail=5
Constraint: alpha < 1.0alpha<1.0.
  ifail = 6ifail=6
On entry, one or more of the off-diagonal elements of h were negative.
  ifail = 7ifail=7
Routine fails to converge in __ iterations.
Increase maxit or check the call to the function.
W ifail = 8ifail=8
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG with details of your call.
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

The returned accuracy is controlled by errtol and limited by machine precision. If eiei is the value of norm_p at the iith iteration, that is
ei = H(GX)F ,
ei = H(G-X)F ,
where HH has been scaled as described above, then the algorithm terminates when:
(|eiei1|)/( 1 + max (ei,ei1) ) errtol .
|ei-ei-1| 1+ max(ei,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 + max (2 × n × n + 6 × n + 1,120 + 9 × n)15×n+5×n×n+max(2×n×n+6×n+1,120+9×n) double elements and 5 × n + 35×n+3 integer elements. All allocated memory is freed before return of nag_nearest_correlation_h_weight (g02aj).

Example

function nag_nearest_correlation_h_weight_example
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] = nag_nearest_correlation_h_weight(g, alpha, h);

if (ifail == 0)
  fprintf('\nReturned H Matrix\n');
  disp(h);
  fprintf('Nearest Correlation Matrix\n');
  disp(x);
  fprintf('Number of iterations taken:     %d\n', iter);
  fprintf('Norm value: %26.4f\n', norm_p);
  fprintf('Alpha:      %26.4f\n', alpha);

  [~, w, info] = nag_lapack_dsyev('n', 'u', x);
  fprintf('\nEigenvalues of X\n');
  disp(w');
end
 

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


function g02aj_example
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);

if (ifail == 0)
  fprintf('\nReturned H Matrix\n');
  disp(h);
  fprintf('Nearest Correlation Matrix\n');
  disp(x);
  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');
  disp(w');
end
 

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



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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