NAG Library Routine Document

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 <nagmk26.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)

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

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

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

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: $g (ldg, n)$ – Real (Kind=nag_wp) arrayInput/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: $ldg$ – IntegerInput: On entry: the first dimension of the array g as declared in the (sub)program from which g02apf is called.

Constraint: $ldg \geq n$ .
3: $n$ – IntegerInput: 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: $h (ldh, n)$ – Real (Kind=nag_wp) arrayInput/Output: 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: $ldh$ – IntegerInput: On entry: the first dimension of the array h as declared in the (sub)program from which g02apf is called.

Constraint: $ldh \geq n$ .
7: $errtol$ – Real (Kind=nag_wp)Input: On entry: the termination tolerance for the iteration.
If $errtol \leq 0$ , $\sqrt{machine precision}$ is used. See Section 7 for further details.
8: $eigtol$ – Real (Kind=nag_wp)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: $x (ldx, n)$ – Real (Kind=nag_wp) arrayOutput: On exit: contains the matrix $X$ .
10: $ldx$ – IntegerInput: On entry: the first dimension of the array x as declared in the (sub)program from which g02apf is called.

Constraint: $ldx \geq n$ .
11: $alpha$ – Real (Kind=nag_wp)Output: On exit: the constant $α$ used in the formation of $X$ .
12: $iter$ – IntegerOutput: 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 ${‖G - X‖}_{F}$ after the final iteration.
15: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . 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

- 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: $ldg \geq n$ .

$ifail = 3$: On entry, $theta = 〈value〉$ .
Constraint: $theta < 1.0$ .

$ifail = 4$: On entry, $ldh = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldh \geq n$ .

$ifail = 5$: On entry, $ldx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldx \geq n$ .

$ifail = 6$: 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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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 (dpotrf).

The number of iterations taken for the bisection will be:

⌈\log_{2} (\frac{1}{errtol})⌉ .

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

real elements. All allocated memory is freed before return of g02apf.

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 Library Routine Document

g02apf (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

1

Purpose