NAG CL Interface
g02anc (corrmat_​shrinking)

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1 Purpose

g02anc computes a correlation matrix, subject to preserving a leading principal submatrix and applying the smallest relative perturbation to the remainder of the approximate input matrix.

2 Specification

#include <nag.h>
void  g02anc (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)
The function may be called by the names: g02anc, nag_correg_corrmat_shrinking or nag_nearest_correlation_shrinking.

3 Description

g02anc finds a correlation matrix, X, starting from an approximate correlation matrix, G, with positive definite leading principal 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×k leading principal submatrix of the input matrix G and positive definite, and α[0,1].
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] double Input/Output
On entry: G, the initial matrix.
On exit: a symmetric matrix 12(G+GT) with the diagonal set to I.
2: pdg Integer Input
On entry: the stride separating row elements of the matrix G in the array g.
Constraint: pdgn.
3: n Integer Input
On entry: the order of the matrix G.
Constraint: n>0.
4: k Integer Input
On entry: k, the order of the leading principal submatrix A.
Constraint: nk>0.
5: errtol double Input
On entry: the termination tolerance for the iteration.
If errtol0.0, machine precision is used. See Section 7 for further details.
6: eigtol double Input
On entry: the tolerance used in determining the definiteness of A.
If λmin(A)>n×λmax(A)×eigtol, where λmin(A) and λmax(A) denote the minimum and maximum eigenvalues of A respectively, A is positive definite.
If eigtol0, machine precision is used.
7: x[pdx×n] double Output
On exit: contains the matrix X.
8: pdx Integer Input
On entry: the stride separating row elements of the matrix X 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 principal 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
The k×k principal 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 8 in the Introduction to the NAG Library CL Interface 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 successfully factorized with a call to f07fdc.
The number of iterations taken for the bisection will be:
log2(1errtol) .  

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g02anc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 g02anc. The total size of these arrays does not exceed 2×n2+3×n real elements. All allocated memory is freed before return of g02anc.

10 Example

This example finds the smallest uniform perturbation α to G, such that the output is a correlation matrix and the k×k leading principal 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)