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>

void	g02apc (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)

The function may be called by the names: g02apc, nag_correg_corrmat_target or nag_nearest_correlation_target.

3 Description

Starting from an approximate correlation matrix,

G

, 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

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: $g [pdg \times n]$ – double Input/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: $pdg$ – Integer Input: On entry: the stride separating row elements of the matrix $G$ in the array g.

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

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

Constraint: $theta < 1.0$ .
5: $h [pdh \times n]$ – double Input/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: $pdh$ – Integer Input: On entry: the stride separating matrix row elements in the array h.

Constraint: $pdh \geq n$ .
7: $errtol$ – double Input: On entry: the termination tolerance for the iteration.
If $errtol \leq 0.0$ , $\sqrt{machine precision}$ is used. See Section 7 for further details.
8: $eigtol$ – double Input: 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: $x [pdx \times n]$ – double Output: On exit: contains the matrix $X$ .
10: $pdx$ – Integer Input: On entry: the stride separating row 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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: $pdg \geq n$ .

On entry, $pdh = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdh \geq n$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq n$ .
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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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 f07fdc.

The number of iterations taken for the bisection will be:

⌈ \log_{2} (\frac{1}{errtol}) ⌉ .

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02apc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 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 g02apc.

10 Example

This example finds the smallest

α

such that

α (H \circ G) + (1 - α) G

is a correlation matrix. The

2 \times 2

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

2 \times 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}) .

g02ap: FL CL CPP AD PY MB

NAG CL Interfaceg02apc (corrmat_​target)

▸▿ 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

NAG CL Interface
g02apc (corrmat_target)