nag_nearest_correlation (g02aac) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_nearest_correlation (g02aac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_nearest_correlation (g02aac) computes the nearest correlation matrix, in the Frobenius norm, to a given square, input matrix.

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_nearest_correlation (Nag_OrderType order, double g[], Integer pdg, Integer n, double errtol, Integer maxits, Integer maxit, double x[], Integer pdx, Integer *iter, Integer *feval, double *nrmgrd, NagError *fail)

3  Description

A correlation matrix may be characterized as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.
nag_nearest_correlation (g02aac) 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).

4  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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     g[dim]doubleInput/Output
Note: the dimension, dim, of the array g must be at least pdg×n.
The i,jth element of the matrix G is stored in
  • g[j-1×pdg+i-1] when order=Nag_ColMajor;
  • g[i-1×pdg+j-1] when order=Nag_RowMajor.
On entry: G, the initial matrix.
On exit: a symmetric matrix 12G+GT with the diagonal set to I.
3:     pdgIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array g.
Constraint: pdgn.
4:     nIntegerInput
On entry: the size of the matrix G.
Constraint: n>0.
5:     errtoldoubleInput
On entry: the termination tolerance for the Newton iteration. If errtol0.0 then n×machine precision is used.
6:     maxitsIntegerInput
On entry: 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, 2×n is used.
7:     maxitIntegerInput
On entry: specifies the maximum number of Newton iterations.
If maxit0, 200 is used.
8:     x[dim]doubleOutput
Note: the dimension, dim, of the array x must be at least pdx×n.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On exit: contains the nearest correlation matrix.
9:     pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraint: pdxn.
10:   iterInteger *Output
On exit: the number of Newton steps taken.
11:   fevalInteger *Output
On exit: the number of function evaluations of the dual problem.
12:   nrmgrddouble *Output
On exit: the norm of the gradient of the last Newton step.
13:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
Machine precision is limiting convergence.
The array returned in x may still be of interest.
Newton iteration fails to converge in value iterations.
NE_EIGENPROBLEM
Failure to solve intermediate eigenproblem.
NE_INT
On entry, n=value.
Constraint: n>0.
NE_INT_2
On entry, pdg=value and n=value.
Constraint: pdgn.
On entry, pdx=value and n=value.
Constraint: pdxn.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

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

8  Further Comments

Arrays are internally allocated by nag_nearest_correlation (g02aac). The total size of these arrays is 11×n+3×n×n+max2×n×n+6×n+1,120+9×n real elements and 5×n+3 integer elements.

9  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

9.1  Program Text

Program Text (g02aace.c)

9.2  Program Data

None.

9.3  Program Results

Program Results (g02aace.r)


nag_nearest_correlation (g02aac) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012