NAG C Library Function Document

nag_nearest_correlation_shrinking (g02anc)


nag_nearest_correlation_shrinking (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.


#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)


nag_nearest_correlation_shrinking (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 by k leading principal 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.


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


1:     g[pdg×n] doubleInput/Output
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 column elements of the matrix G 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 principal 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
On exit: contains the matrix X.
8:     pdx IntegerInput
On entry: the stride separating column 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 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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: pdg or n.
On entry, pdx=value and n=value.
Constraint: pdx or n.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
The k by 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.


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 nag_dpotrf (f07fdc).
The number of iterations taken for the bisection will be:
log21errtol .  

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.

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).


This example finds the smallest uniform perturbation α to G, such that the output is a correlation matrix and the k by 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 .  

Program Text

Program Text (g02ance.c)

Program Data

Program Data (g02ance.d)

Program Results

Program Results (g02ance.r)