NAG C Library Function Document

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,

G

, nag_nearest_correlation_target (g02apc) finds a correlation matrix,

X

, which has the form

X = α T + (1 - α) G,

where

α \in [0, 1]

and

T = H \circ G

is a target matrix.

C = A \circ B

denotes the matrix

C

with elements

C_{i j} = A_{i j} \times B_{i j}

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

H

essentially fixes an element in

G

so it is unchanged in

X

nag_nearest_correlation_target (g02apc) 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

θ \in [0, 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: $g [pdg \times n]$ – doubleInput/Output: On entry: $G$ , the initial matrix.

On exit: a symmetric matrix $\frac{1}{2} (G + G^{T})$ with the diagonal elements set to $1.0$ .
2: $pdg$ – IntegerInput: On entry: the stride separating column elements of the matrix $G$ in the array g.

Constraint: $pdg \geq n$ .
3: $n$ – IntegerInput: On entry: the order of the matrix $G$ .

Constraint: $n > 0$ .
4: $theta$ – doubleInput: On entry: the value of $θ$ . If $theta < 0.0$ , $0.0$ is used.

Constraint: $theta < 1.0$ .
5: $h [pdh \times n]$ – doubleInput/Output: Note: the $(i, j)$ th element of the matrix $H$ is stored in $h [(j - 1) \times pdh + i - 1]$ .

On entry: the matrix of weights $H$ .

On exit: a symmetric matrix $\frac{1}{2} (H + H^{T})$ with its diagonal elements set to $1.0$ .
6: $pdh$ – IntegerInput: On entry: the stride separating matrix row elements in the array h.

Constraint: $pdh \geq n$ .
7: $errtol$ – doubleInput: On entry: the termination tolerance for the iteration.
If $errtol \leq 0$ , $\sqrt{machine precision}$ is used. See Section 7 for further details.
8: $eigtol$ – doubleInput: On entry: the tolerance used in determining the definiteness of the target matrix $T = H \circ G$ .
If $λ_{\min} (T) > n \times λ_{\max} (T) \times eigtol$ , where $λ_{\min} (T)$ and $λ_{\max} (T)$ denote the minimum and maximum eigenvalues of $T$ respectively, $T$ is positive definite.

If $eigtol \leq 0$ , machine precision is used.
9: $x [pdx \times n]$ – doubleOutput: On exit: contains the matrix $X$ .
10: $pdx$ – IntegerInput: On entry: the stride separating column elements of the matrix $X$ in the array x.

Constraint: $pdx \geq n$ .
11: $alpha$ – double *Output: On exit: the constant $α$ used in the formation of $X$ .
12: $iter$ – Integer *Output: On exit: the number of iterations taken.
13: $eigmin$ – double *Output: On exit: the smallest eigenvalue of the target matrix $T$ .
14: $norm$ – double *Output: On exit: the value of ${‖G - X‖}_{F}$ after the final iteration.
15: $fail$ – 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 $〈value〉$ had an illegal value.
NE_EIGENPROBLEM: Failure to solve intermediate eigenproblem. This should not occur. Please contact NAG.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .
NE_INT_2: On entry, $pdg = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdg \geq$ or $n$ .

On entry, $pdh = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdh \geq$ or $n$ .

On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq$ or $n$ .
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, $theta = 〈value〉$ .
Constraint: $theta < 1.0$ .

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 nag_dpotrf (f07fdc).

The number of iterations taken for the bisection will be:

⌈\log_{2} (\frac{1}{errtol})⌉ .

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.

9

Further Comments

Arrays are internally allocated by nag_nearest_correlation_target (g02apc). The total size of these arrays does not exceed

2 \times n^{2} + 3 \times n

real elements. All allocated memory is freed before return of nag_nearest_correlation_target (g02apc).

10

Example

This example finds the smallest

α

such that

α (H \circ G) + (1 - α) G

is a correlation matrix. The

2

2

leading principal submatrix of the input is preserved, and the last

2

2

diagonal block is weighted to give some emphasis to the off diagonal elements.

G = (\begin{array}{r} 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 \end{array})

and

H = (\begin{matrix} 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 \end{matrix}) .

NAG C Library Function Document

nag_nearest_correlation_target (g02apc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose