NAG FL Interface
g02aaf (corrmat_nearest)

Integer, Intent (In)	::	ldg, n, maxits, maxit, ldx
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	iter, feval
Real (Kind=nag_wp), Intent (In)	::	errtol
Real (Kind=nag_wp), Intent (Inout)	::	g(ldg,n), x(ldx,n)
Real (Kind=nag_wp), Intent (Out)	::	nrmgrd

C Header Interface

#include <nag.h>

void	g02aaf_ (double g[], const Integer ldg, const Integer n, const double errtol, const Integer maxits, const Integer maxit, double x[], const Integer ldx, Integer iter, Integer feval, double nrmgrd, Integer ifail)

The routine may be called by the names g02aaf or nagf_correg_corrmat_nearest.

3 Description

A correlation matrix may be characterised as a real square matrix that is symmetric, has a unit diagonal and is positive semidefinite.

g02aaf applies an inexact Newton method to a dual formulation of the problem, as described by Qi and Sun (2006). It applies the improvements suggested by Borsdorf and Higham (2010).

4 References

Borsdorf R and Higham N J (2010) A preconditioned (Newton) algorithm for the nearest correlation matrix IMA Journal of Numerical Analysis 30(1) 94–107

Qi H and Sun D (2006) A quadratically convergent Newton method for computing the nearest correlation matrix SIAM J. Matrix AnalAppl 29(2) 360–385

5 Arguments

1: $g (ldg, n)$ – Real (Kind=nag_wp) array Input/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: $ldg$ – Integer Input: On entry: the first dimension of the array g as declared in the (sub)program from which g02aaf is called.

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

Constraint: $n > 0$ .
4: $errtol$ – Real (Kind=nag_wp) Input: On entry: the termination tolerance for the Newton iteration. If $errtol \leq 0.0$ , $n \times \sqrt{machine precision}$ is used.
5: $maxits$ – Integer Input: On entry: maxits specifies the maximum number of iterations used for the iterative scheme used to solve the linear algebraic equations at each Newton step.
If $maxits \leq 0$ , $100$ is used.
6: $maxit$ – Integer Input: On entry: specifies the maximum number of Newton iterations.
If $maxit \leq 0$ , $200$ is used.
7: $x (ldx, n)$ – Real (Kind=nag_wp) array Output: On exit: contains the nearest correlation matrix.
8: $ldx$ – Integer Input: On entry: the first dimension of the array x as declared in the (sub)program from which g02aaf is called.

Constraint: $ldx \geq n$ .
9: $iter$ – Integer Output: On exit: the number of Newton steps taken.
10: $feval$ – Integer Output: On exit: the number of function evaluations of the dual problem.
11: $nrmgrd$ – Real (Kind=nag_wp) Output: On exit: the norm of the gradient of the last Newton step.
12: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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, $ldg = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldg \geq n$ .

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

On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

$ifail = 2$: Newton iteration fails to converge in $⟨ value ⟩$ iterations.

$ifail = 3$: Machine precision is limiting convergence.
The array returned in x may still be of interest.

$ifail = 4$: An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The returned accuracy is controlled by errtol and limited by machine precision.

8 Parallelism and Performance

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

g02aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02aaf 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 g02aaf. The total size of these arrays is

11 \times n + 3 \times n \times n + \max (2 \times n \times n + 6 \times n + 1, 120 + 9 \times n)

real elements and

5 \times n + 3

integer elements.

It is important to note that the nearest correlation matrix, to the symmetrized matrix supplied, may not be positive definite. That is, at least one of the eigenvalues of the returned correlation matrix may be zero relative to the accuracy requested. Numerically, in such cases, computing the eigenvalues of the returned correlation matrix may yield small and negative eigenvalues; reducing the tolerance used will reduce the size (closer to zero) of such eigenvalues, but some computed eigenvalues may remain negative.

To ensure a correlation matrix that is positive definite with small positive eigenvalues, you can call f07fdf on a copy of the returned correlation matrix (

C = x

, say). If

info > 0

then the leading matrix of order

r = info - 1

of x is positive definite. x and

r

can then be passed as arguments g and k to g02anf to obtain a positive definite correlation matrix (with some very small positive eigenvalues) that is very close to the matrix returned by g02aaf.

Alternatively, a correlation matrix with zero eigenvalues (to the accuracy required) suggests that some (

n - r

) of the correlated variables are redundant and should be considered for removal from the data model.

10 Example

This example finds the nearest correlation matrix to the matrix

G

, where

G = (\begin{matrix} 2 & −1 & 0 & 0 \\ −1 & 2 & −1 & 0 \\ 0 & −1 & 2 & −1 \\ 0 & 0 & −1 & 2 \end{matrix}) .

The example also demonstrates how a modified Cholesky factorization (computed using f01mdf and f01mef) can be used to obtain a bound on the distance to the nearest correlation matrix prior to computing the NCM itself.

NAG FL Interfaceg02aaf (corrmat_​nearest)

▸▿ Contents

1 Purpose

2 Specification