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).

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: $order$ – 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 2.3.1.3 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

Nag_ColMajor

2: $g [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array g must be at least

pdg \times n

The

(i, j)

th element of the matrix

G

is stored in

$g [(j - 1) \times pdg + i - 1]$ when $order = Nag_ColMajor$ ;
$g [(i - 1) \times pdg + j - 1]$ when $order = Nag_RowMajor$ .

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) 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.

maxits \leq 0

2 \times n

is used.

7: $maxit$ – IntegerInput

On entry: specifies the maximum number of Newton iterations.

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

The

(i, j)

th element of the matrix

X

is stored in

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

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) 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 2.7 in How to Use the NAG Library and its Documentation).

6 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: $n > 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
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.

7 Accuracy

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

8 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.