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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:
Mark 23: errtol, maxits, maxit now 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:     g(ldg,n) – double array
ldg, the first dimension of the array, must satisfy the constraint ldgn ldgn .
G G , the initial matrix.

Optional Input Parameters

1:     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 G .
Constraint: n > 0 n>0 .
2:     errtol – double scalar
The termination tolerance for the Newton iteration. If errtol0.0 errtol0.0  then n × sqrt(machine precision) n×machine precision  is used.
Default: 0.0 0.0  
3:     maxits – int64int32nag_int scalar
maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If maxits0 maxits0 , 2 × n 2×n  is used.
Default: 0 0  
4:     maxit – int64int32nag_int scalar
Specifies the maximum number of Newton iterations.
If maxit0 maxit0 , 200 200  is used.
Default: 0 0  

Input Parameters Omitted from the MATLAB Interface

ldg ldx

Output Parameters

1:     g(ldg,n) – double array
ldgn ldgn .
A symmetric matrix (1/2)(G + GT) 12 (G+GT) with the diagonal set to I I .
2:     x(ldx,n) – double array
ldxn ldxn .
Contains the nearest correlation matrix.
3:     iter – int64int32nag_int scalar
The number of Newton steps taken.
4:     feval – int64int32nag_int scalar
The number of function evaluations of the dual problem.
5:     nrmgrd – double scalar
The norm of the gradient of the last Newton step.
6:     ifail – int64int32nag_int scalar
ifail = 0 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 ifail=1
On entry,n0 n0 ,
orldg < n ldg<n ,
orldx < n ldx<n .
  ifail = 2 ifail=2
The function fails to converge in maxit iterations. Increase maxit or check the call to the function.
W ifail = 3 ifail=3
Machine precision is limiting convergence. In this instance the returned value of x may be useful.
  ifail = 4 ifail=4
An internal eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
  ifail = 999 ifail=-999
Internal 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 (g02aa). The total size of these arrays is 11 × n + 3 × n × n + max (2 × n × n + 6 × n + 1,120 + 9 × n) 11×n+3×n×n+max (2×n×n+6×n+1,120+9×n)  real elements and 5 × n + 3 5×n+3  integer elements.

Example

function nag_correg_corrmat_nearest_example
g = [2, -1, 0, 0;
     -1, 2, -1, 0;
     0, -1, 2, -1;
     0, 0, -1, 2];
[gOut, x, iter, feval, nrmgrd, ifail] = nag_correg_corrmat_nearest(g);
if (ifail == 0)
  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*nag_machine_precision)
    fprintf(' Norm of gradient of last Newton step: %6.4f\n', nrmgrd);
  end
end
 

     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

function g02aa_example
g = [2, -1, 0, 0;
     -1, 2, -1, 0;
     0, -1, 2, -1;
     0, 0, -1, 2];
[gOut, x, iter, feval, nrmgrd, ifail] = g02aa(g);
if (ifail == 0)
  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
end
 

     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


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Chapter Introduction
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