nag_nearest_correlation_shrinking (g02anc) (PDF version)
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g02 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_nearest_correlation_shrinking (g02anc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_nearest_correlation_shrinking (g02anc) computes a correlation matrix, subject to preserving a leading principle submatrix and applying the smallest uniform perturbation to the remainder of the approximate input matrix.

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_nearest_correlation_shrinking (double g[], Integer pdg, Integer n, Integer k, double errtol, double eigtol, double x[], Integer pdx, double *alpha, Integer *iter, double *eigmin, double *norm, NagError *fail)

3  Description

nag_nearest_correlation_shrinking (g02anc) finds a correlation matrix, X, starting from an approximate correlation matrix, G, with positive definite leading principle submatrix of order k. The returned correlation matrix, X, has the following structure:
X = α A 0 0 I + 1-α G  
where A is the k by k leading principle submatrix of the input matrix G and positive definite, and α0,1.
nag_nearest_correlation_shrinking (g02anc) utilizes a shrinking method to find the minimum value of α such that X is positive definite with unit diagonal.

4  References

Higham N J, Strabić N and Šego V (2014) Restoring definiteness via shrinking, with an application to correlation matrices with a fixed block MIMS EPrint 2014.54 Manchester Institute for Mathematical Sciences, The University of Manchester, UK

5  Arguments

1:     g[pdg×n] doubleInput/Output
Note: the i,jth element of the matrix G is stored in g[j-1×pdg+i-1].
On entry: G, the initial matrix.
On exit: a symmetric matrix 12G+GT with the diagonal set to I.
2:     pdg IntegerInput
On entry: the stride separating matrix row elements in the array g.
Constraint: pdgn.
3:     n IntegerInput
On entry: the order of the matrix G.
Constraint: n>0.
4:     k IntegerInput
On entry: k, the order of the leading principle submatrix A.
Constraint: nk>0.
5:     errtol doubleInput
On entry: the termination tolerance for the iteration.
If errtol0, machine precision is used. See Section 7 for further details.
6:     eigtol doubleInput
On entry: the tolerance used in determining the definiteness of A.
If λminA>n×λmaxA×eigtol, where λminA and λmaxA denote the minimum and maximum eigenvalues of A respectively, A is positive definite.
If eigtol0, machine precision is used.
7:     x[pdx×n] doubleOutput
Note: the i,jth element of the matrix X is stored in x[j-1×pdx+i-1].
On exit: contains the matrix X.
8:     pdx IntegerInput
On entry: the stride separating matrix row elements in the array x.
Constraint: pdxn.
9:     alpha double *Output
On exit: α.
10:   iter Integer *Output
On exit: the number of iterations taken.
11:   eigmin double *Output
On exit: the smallest eigenvalue of the leading principle submatrix A.
12:   norm double *Output
On exit: the value of G-XF after the final iteration.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in the Essential Introduction for further information.
On entry, argument value had an illegal value.
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG.
On entry, n=value.
Constraint: n>0.
On entry, k=value and n=value.
Constraint: nk>0.
On entry, pdg=value and n=value.
Constraint: pdgn.
On entry, pdx=value and n=value.
Constraint: pdxn.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
The k-by-k principle leading submatrix of the initial matrix G is not positive definite.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

7  Accuracy

The algorithm uses a bisection method. It is terminated when the computed α is within errtol of the minimum value. The positive definiteness of X is such that it can be sucessfully factorized with a call to nag_dpotrf (f07fdc).
The number of interations taken for the bisection will be:
log21errtol .  

8  Parallelism and Performance

nag_nearest_correlation_shrinking (g02anc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_shrinking (g02anc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

Arrays are internally allocated by nag_nearest_correlation_shrinking (g02anc). The total size of these arrays does not exceed 2×n2+3×n real elements. All allocated memory is freed before return of nag_nearest_correlation_shrinking (g02anc).

10  Example

This example finds the smallest uniform perturbation α to G, such that the output is a correlation matrix and the k-by-k leading principle submatrix of the input is preserved,
G = 1.0000 -0.0991 0.5665 -0.5653 -0.3441 -0.0991 1.0000 -0.4273 0.8474 0.4975 0.5665 -0.4273 1.0000 -0.1837 -0.0585 -0.5653 0.8474 -0.1837 1.0000 -0.2713 -0.3441 0.4975 -0.0585 -0.2713 1.0000 .  

10.1  Program Text

Program Text (g02ance.c)

10.2  Program Data

Program Data (g02ance.d)

10.3  Program Results

Program Results (g02ance.r)

nag_nearest_correlation_shrinking (g02anc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015