void	nag_nearest_correlation (Nag_OrderType order, double g[], Integer pdg, Integer n, double errtol, Integer maxits, Integer maxit, double x[], Integer pdx, Integer iter, Integer feval, double nrmgrd, NagError fail)

3

Description

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

nag_nearest_correlation (g02aac) 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).

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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

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Arguments

1: $order$ – Nag_OrderTypeInput: On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $order = Nag_RowMajor$ . See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint: $order = Nag_RowMajor$ or $Nag_ColMajor$ .
2: $g [\dim]$ – doubleInput/Output: Note: the dimension, dim, of the array g must be at least $pdg \times n$ .

On entry: $G$ , the initial matrix.

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

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

Constraint: $n > 0$ .
5: $errtol$ – doubleInput: On entry: the termination tolerance for the Newton iteration. If $errtol \leq 0.0$ then $n \times \sqrt{machine precision}$ is used.
6: $maxits$ – IntegerInput: 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.
7: $maxit$ – IntegerInput: On entry: specifies the maximum number of Newton iterations.
If $maxit \leq 0$ , $200$ is used.
8: $x [\dim]$ – doubleOutput: Note: the dimension, dim, of the array x must be at least $pdx \times n$ .

On exit: contains the nearest correlation matrix.
9: $pdx$ – IntegerInput: On entry: the stride separating row or column elements (depending on the value of order) of the matrix $X$ in the array x.

Constraint: $pdx \geq n$ .
10: $iter$ – Integer *Output: On exit: the number of Newton steps taken.
11: $feval$ – Integer *Output: On exit: the number of function evaluations of the dual problem.
12: $nrmgrd$ – double *Output: On exit: the norm of the gradient of the last Newton step.
13: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

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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_CONVERGENCE: Machine precision is limiting convergence.
The array returned in x may still be of interest.

Newton iteration fails to converge in $〈value〉$ iterations.
NE_EIGENPROBLEM: An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .

On entry, $pdg = 〈value〉$ .
Constraint: $pdg > 0$ .

On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pdg = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdg \geq n$ .

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.

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Accuracy

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

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Parallelism and Performance

nag_nearest_correlation (g02aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_nearest_correlation (g02aac) 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.

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Further Comments

Arrays are internally allocated by nag_nearest_correlation (g02aac). 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.

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Example

This example finds the nearest correlation matrix to:

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

NAG Library Function Document

nag_nearest_correlation (g02aac)

▸▿ 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

2

Specification