NAG Library Routine Document

G02ANF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G02ANF computes a correlation matrix, subject to preserving a leading principle submatrix and applying the smallest uniform perturbation to the remainder of the approximate input matrix.

2 Specification

SUBROUTINE G02ANF (

G, LDG, N, K, ERRTOL, EIGTOL, X, LDX, ALPHA, ITER, EIGMIN, NORM, IFAIL)

INTEGER	LDG, N, K, LDX, ITER, IFAIL
REAL (KIND=nag_wp)	G(LDG,N), ERRTOL, EIGTOL, X(LDX,N), ALPHA, EIGMIN, NORM

3 Description

G02ANF finds a correlation matrix,

X

, starting from an approximate correlation matrix,

G

, with positive definite leading principle submatrix of order

k

. The returned correlation matrix,

X

, has the following structure:

X = α (\begin{matrix} A & 0 \\ 0 & I \end{matrix}) + (1 - α) G

where

A

is the

k

k

leading principle submatrix of the input matrix

G

and positive definite, and

α \in [0, 1]

G02ANF utilizes a shrinking method to find the minimum value of

α

such that

X

is positive definite with unit diagonal.

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 Parameters

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 set to $I$ .
2: $LDG$ – INTEGERInput: On entry: the first dimension of the array G as declared in the (sub)program from which G02ANF is called.

Constraint: $LDG \geq N$ .
3: $N$ – INTEGERInput: On entry: the order of the matrix $G$ .

Constraint: $N > 0$ .
4: $K$ – INTEGERInput: On entry: $k$ , the order of the leading principle submatrix $A$ .

Constraint: $N \geq K > 0$ .
5: $ERRTOL$ – 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.
6: $EIGTOL$ – REAL (KIND=nag_wp)Input: On entry: the tolerance used in determining the definiteness of $A$ .
If $λ_{\min} (A) > N \times λ_{\max} (A) \times EIGTOL$ , where $λ_{\min} (A)$ and $λ_{\max} (A)$ denote the minimum and maximum eigenvalues of $A$ respectively, $A$ is positive definite.

If $EIGTOL \leq 0$ , machine precision is used.
7: $X (LDX, N)$ – REAL (KIND=nag_wp) arrayOutput: On exit: contains the matrix $X$ .
8: $LDX$ – INTEGERInput: On entry: the first dimension of the array X as declared in the (sub)program from which G02ANF is called.

Constraint: $LDX \geq N$ .
9: $ALPHA$ – REAL (KIND=nag_wp)Output: On exit: $α$ .
10: $ITER$ – INTEGEROutput: On exit: the number of iterations taken.
11: $EIGMIN$ – REAL (KIND=nag_wp)Output: On exit: the smallest eigenvalue of the leading principle submatrix $A$ .
12: $NORM$ – REAL (KIND=nag_wp)Output: On exit: the value of ${‖G - X‖}_{F}$ after the final iteration.
13: $IFAIL$ – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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, $K = 〈value〉$ and $N = 〈value〉$ .
Constraint: $N \geq K > 0$ .

$IFAIL = 4$: On entry, $LDX = 〈value〉$ and $N = 〈value〉$ .
Constraint: $LDX \geq N$ .

$IFAIL = 5$: The $k$ -by- $k$ principle leading submatrix of the initial matrix $G$ is not positive definite.

$IFAIL = 6$: 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 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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. The positive definiteness of

X

is such that it can be sucessfully factorized with a call to F07FDF (DPOTRF).

The number of interations taken for the bisection will be:

⌈\log_{2} (\frac{1}{ERRTOL})⌉ .

8 Parallelism and Performance

G02ANF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G02ANF 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 G02ANF. 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 G02ANF.

10 Example

This example finds the smallest uniform perturbation

α

G

, such that the output is a correlation matrix and the

k

-by-

k

leading principle submatrix of the input is preserved,

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}) .

NAG Library Routine DocumentG02ANF