NAG FL Interface
g02anf (corrmat_​shrinking)

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1 Purpose

g02anf 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

Fortran Interface
Subroutine g02anf ( g, ldg, n, k, errtol, eigtol, x, ldx, alpha, iter, eigmin, norm, ifail)
Integer, Intent (In) :: ldg, n, k, ldx
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iter
Real (Kind=nag_wp), Intent (In) :: errtol, eigtol
Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), x(ldx,n)
Real (Kind=nag_wp), Intent (Out) :: alpha, eigmin, norm
C Header Interface
#include <nag.h>
void  g02anf_ (double g[], const Integer *ldg, const Integer *n, const Integer *k, const double *errtol, const double *eigtol, double x[], const Integer *ldx, double *alpha, Integer *iter, double *eigmin, double *norm, Integer *ifail)
The routine may be called by the names g02anf or nagf_correg_corrmat_shrinking.

3 Description

g02anf 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].
g02anf 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(ldg,n) Real (Kind=nag_wp) array Input/Output
On entry: G, the initial matrix.
On exit: a symmetric matrix 12(G+GT) with the diagonal set to I.
2: ldg Integer Input
On entry: the first dimension of the array g as declared in the (sub)program from which g02anf is called.
Constraint: ldgn.
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 Real (Kind=nag_wp) Input
On entry: the termination tolerance for the iteration.
If errtol0.0, machine precision is used. See Section 7 for further details.
6: eigtol Real (Kind=nag_wp) 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(ldx,n) Real (Kind=nag_wp) array Output
On exit: contains the matrix X.
8: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02anf is called.
Constraint: ldxn.
9: alpha Real (Kind=nag_wp) Output
On exit: α.
10: iter Integer Output
On exit: the number of iterations taken.
11: eigmin Real (Kind=nag_wp) Output
On exit: the smallest eigenvalue of the leading principal submatrix A.
12: norm Real (Kind=nag_wp) Output
On exit: the value of G-XF after the final iteration.
13: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n>0.
On entry, ldg=value and n=value.
Constraint: ldgn.
On entry, k=value and n=value.
Constraint: nk>0.
On entry, ldx=value and n=value.
Constraint: ldxn.
The k×k principal leading submatrix of the initial matrix G is not positive definite.
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL 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 f07fdf.
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.
g02anf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02anf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Arrays are internally allocated by g02anf. The total size of these arrays does not exceed 2×n2+3×n real elements. All allocated memory is freed before return of g02anf.

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 (g02anfe.f90)

10.2 Program Data

Program Data (g02anfe.d)

10.3 Program Results

Program Results (g02anfe.r)