NAG Library Function Document
nag_nearest_correlation_target (g02apc)
1
Purpose
nag_nearest_correlation_target (g02apc) 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
#include <nag.h> |
#include <nagg02.h> |
void |
nag_nearest_correlation_target (double g[],
Integer pdg,
Integer n,
double theta,
double h[],
Integer pdh,
double errtol,
double eigtol,
double x[],
Integer pdx,
double *alpha,
Integer *iter,
double *eigmin,
double *norm,
NagError *fail) |
|
3
Description
Starting from an approximate correlation matrix,
,
nag_nearest_correlation_target (g02apc) finds a correlation matrix,
, which has the form
where
and
is a target matrix.
denotes the matrix
with elements
.
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
in magnitude. A value of
in
essentially fixes an element in
so it is unchanged in
.
nag_nearest_correlation_target (g02apc) utilizes a shrinking method to find the minimum value of such that is positive definite with unit diagonal and with a smallest eigenvalue of at least 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:
– doubleInput/Output
-
On entry: , the initial matrix.
On exit: a symmetric matrix with the diagonal elements set to .
- 2:
– IntegerInput
On entry: the stride separating column elements of the matrix
in the array
g.
Constraint:
.
- 3:
– IntegerInput
-
On entry: the order of the matrix .
Constraint:
.
- 4:
– doubleInput
-
On entry: the value of . If , is used.
Constraint:
.
- 5:
– doubleInput/Output
-
Note: the th element of the matrix is stored in .
On entry: the matrix of weights .
On exit: a symmetric matrix with its diagonal elements set to .
- 6:
– IntegerInput
-
On entry: the stride separating matrix row elements in the array
h.
Constraint:
.
- 7:
– doubleInput
-
On entry: the termination tolerance for the iteration.
If
,
is used. See
Section 7 for further details.
- 8:
– doubleInput
-
On entry: the tolerance used in determining the definiteness of the target matrix
.
If , where and denote the minimum and maximum eigenvalues of respectively, is positive definite.
If , machine precision is used.
- 9:
– doubleOutput
-
On exit: contains the matrix .
- 10:
– IntegerInput
On entry: the stride separating column elements of the matrix
in the array
x.
Constraint:
.
- 11:
– double *Output
-
On exit: the constant used in the formation of .
- 12:
– Integer *Output
-
On exit: the number of iterations taken.
- 13:
– double *Output
-
On exit: the smallest eigenvalue of the target matrix .
- 14:
– double *Output
-
On exit: the value of after the final iteration.
- 15:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_EIGENPROBLEM
-
Failure to solve intermediate eigenproblem. This should not occur. Please contact
NAG.
- NE_INT
-
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
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.
- NE_MAT_NOT_POS_DEF
-
The target matrix is not positive definite.
- NE_NO_LICENCE
-
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.
- NE_REAL
-
On entry, .
Constraint: .
7
Accuracy
The algorithm uses a bisection method. It is terminated when the computed
is within
errtol of the minimum value.
Note: when
is zero
is still positive definite, in that it can be successfully factorized with a call to
nag_dpotrf (f07fdc).
The number of iterations taken for the bisection will be:
8
Parallelism and Performance
nag_nearest_correlation_target (g02apc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nearest_correlation_target (g02apc) 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.
Arrays are internally allocated by nag_nearest_correlation_target (g02apc). The total size of these arrays does not exceed real elements. All allocated memory is freed before return of nag_nearest_correlation_target (g02apc).
10
Example
This example finds the smallest
such that
is a correlation matrix. The
by
leading principal submatrix of the input is preserved, and the last
by
diagonal block is weighted to give some emphasis to the off diagonal elements.
and
10.1
Program Text
Program Text (g02apce.c)
10.2
Program Data
Program Data (g02apce.d)
10.3
Program Results
Program Results (g02apce.r)