NAG FL Interface
g02apf (corrmat_​target)

1 Purpose

g02apf computes a correlation matrix, by using a positive definite target matrix derived from weighting the approximate input matrix, with an optional bound on the minimum eigenvalue.

2 Specification

Fortran Interface
Subroutine g02apf ( g, ldg, n, theta, h, ldh, errtol, eigtol, x, ldx, alpha, iter, eigmin, norm, ifail)
Integer, Intent (In) :: ldg, n, ldh, ldx
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iter
Real (Kind=nag_wp), Intent (In) :: theta, errtol, eigtol
Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), h(ldh,n), x(ldx,n)
Real (Kind=nag_wp), Intent (Out) :: alpha, eigmin, norm
C Header Interface
#include <nag.h>
void  g02apf_ (double g[], const Integer *ldg, const Integer *n, const double *theta, double h[], const Integer *ldh, 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 g02apf or nagf_correg_corrmat_target.

3 Description

Starting from an approximate correlation matrix, G, g02apf finds a correlation matrix, X, which has the form
X = α T + 1-α G ,  
where α0,1 and T=HG is a target matrix. C=AB denotes the matrix C with elements Cij=Aij×Bij. H is a matrix of weights that defines the target matrix. The target matrix must be positive definite and thus have off-diagonal elements less than 1 in magnitude. A value of 1 in H essentially fixes an element in G so it is unchanged in X.
g02apf utilizes a shrinking method to find the minimum value of α such that X is positive definite with unit diagonal and with a smallest eigenvalue of at least θ0,1 times the smallest eigenvalue of the target matrix.

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: gldgn Real (Kind=nag_wp) array Input/Output
On entry: G, the initial matrix.
On exit: a symmetric matrix 12G+GT with the diagonal elements set to 1.0.
2: ldg Integer Input
On entry: the first dimension of the array g as declared in the (sub)program from which g02apf is called.
Constraint: ldgn.
3: n Integer Input
On entry: the order of the matrix G.
Constraint: n>0.
4: theta Real (Kind=nag_wp) Input
On entry: the value of θ. If theta<0.0, 0.0 is used.
Constraint: theta<1.0.
5: hldhn Real (Kind=nag_wp) array Input/Output
On entry: the matrix of weights H.
On exit: a symmetric matrix 12H+HT with its diagonal elements set to 1.0.
6: ldh Integer Input
On entry: the first dimension of the array h as declared in the (sub)program from which g02apf is called.
Constraint: ldhn.
7: 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.
8: eigtol Real (Kind=nag_wp) Input
On entry: the tolerance used in determining the definiteness of the target matrix T=HG.
If λminT>n×λmaxT×eigtol, where λminT and λmaxT denote the minimum and maximum eigenvalues of T respectively, T is positive definite.
If eigtol0, machine precision is used.
9: xldxn Real (Kind=nag_wp) array Output
On exit: contains the matrix X.
10: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02apf is called.
Constraint: ldxn.
11: alpha Real (Kind=nag_wp) Output
On exit: the constant α used in the formation of X.
12: iter Integer Output
On exit: the number of iterations taken.
13: eigmin Real (Kind=nag_wp) Output
On exit: the smallest eigenvalue of the target matrix T.
14: norm Real (Kind=nag_wp) Output
On exit: the value of G-XF after the final iteration.
15: 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:
ifail=1
On entry, n=value.
Constraint: n>0.
ifail=2
On entry, ldg=value and n=value.
Constraint: ldgn.
ifail=3
On entry, theta=value.
Constraint: theta<1.0.
ifail=4
On entry, ldh=value and n=value.
Constraint: ldhn.
ifail=5
On entry, ldx=value and n=value.
Constraint: ldxn.
ifail=6
The target matrix is not positive definite.
ifail=7
Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG.
ifail=-99
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.
ifail=-399
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.
ifail=-999
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.
Note: when θ is zero X is still positive definite, in that it can be successfully factorized with a call to f07fdf.
The number of iterations taken for the bisection will be:
log21errtol .  

8 Parallelism and Performance

g02apf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02apf 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 g02apf. The total size of these arrays does not exceed 2×n2+3×n real elements. All allocated memory is freed before return of g02apf.

10 Example

This example finds the smallest α such that αHG+1-αG is a correlation matrix. The 2 by 2 leading principal submatrix of the input is preserved, and the last 2 by 2 diagonal block is weighted to give some emphasis to the off diagonal elements.
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  
and
H = 1.0000 1.0000 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.5000 0.0000 0.0000 0.0000 0.5000 1.0000 .  

10.1 Program Text

Program Text (g02apfe.f90)

10.2 Program Data

Program Data (g02apfe.d)

10.3 Program Results

Program Results (g02apfe.r)