g02abc computes the nearest correlation matrix, in the Frobenius norm or weighted Frobenius norm, and optionally with bounds on the eigenvalues, to a given square, input matrix.

2 Specification

#include <nag.h>

void	g02abc (Nag_OrderType order, double g[], Integer pdg, Integer n, Nag_NearCorr_ProbType opt, double alpha, double w[], double errtol, Integer maxits, Integer maxit, double x[], Integer pdx, Integer iter, Integer feval, double nrmgrd, NagError fail)

The function may be called by the names: g02abc, nag_correg_corrmat_nearest_bounded or nag_nearest_correlation_bounded.

3 Description

Finds the nearest correlation matrix

X

by minimizing

\frac{1}{2} {‖ G - X ‖}^{2}

where

G

is an approximate correlation matrix.

The norm can either be the Frobenius norm or the weighted Frobenius norm

\frac{1}{2} {‖ W^{\frac{1}{2}} (G - X) W^{\frac{1}{2}} ‖}_{F}^{2}

You can optionally specify a lower bound on the eigenvalues,

α

, of the computed correlation matrix, forcing the matrix to be positive definite,

0 < α < 1

Note that if the weights vary by several orders of magnitude from one another the algorithm may fail to converge.

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_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $g [\dim]$ – double Input/Output

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

pdg \times n

On entry:

G

, the initial matrix.

On exit:

G

is overwritten.

3: $pdg$ – Integer Input

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: $n$ – Integer Input

On entry: the order of the matrix

G

Constraint:

n > 0

5: $opt$ – Nag_NearCorr_ProbType Input

On entry: indicates the problem to be solved.

$opt = Nag_LowerBound$: The lower bound problem is solved.
$opt = Nag_WeightedNorm$: The weighted norm problem is solved.
$opt = Nag_Both$: Both problems are solved.

Constraint:

opt = Nag_LowerBound

Nag_WeightedNorm

Nag_Both

6: $alpha$ – double Input

On entry: the value of

α

opt = Nag_WeightedNorm

, alpha need not be set.

Constraint:

0.0 < alpha < 1.0

7: $w [\dim]$ – double Input/Output

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

$n$ when $opt \neq Nag_LowerBound$ ;
otherwise w may be NULL.

On entry: the square roots of the diagonal elements of

W

, that is the diagonal of

W^{\frac{1}{2}}

opt = Nag_LowerBound

, w is not referenced and may be NULL.

On exit: if

opt = Nag_WeightedNorm

Nag_Both

, the array is scaled so

0 < w [i - 1] \leq 1

, for

i = 1, 2, \dots, n

Constraint:

w [i - 1] > 0.0

, for

i = 1, 2, \dots, n

8: $errtol$ – double Input

On entry: the termination tolerance for the Newton iteration. If

errtol \leq 0.0

n \times \sqrt{machine precision}

is used.

9: $maxits$ – Integer Input

On entry: specifies the maximum number of iterations to be used by the iterative scheme to solve the linear algebraic equations at each Newton step.

maxits \leq 0

2 \times n

is used.

10: $maxit$ – Integer Input

On entry: specifies the maximum number of Newton iterations.

maxit \leq 0

200

is used.

11: $x [\dim]$ – double Output

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

pdx \times n

On exit: contains the nearest correlation matrix.

12: $pdx$ – Integer Input

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

13: $iter$ – Integer * Output

On exit: the number of Newton steps taken.

14: $feval$ – Integer * Output

On exit: the number of function evaluations of the dual problem.

15: $nrmgrd$ – double * Output

On exit: the norm of the gradient of the last Newton step.

16: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: Newton iteration fails to converge in $⟨ value ⟩$ iterations. Increase maxit or check the call to the function.

The machine precision is limiting convergence. In this instance the returned value of x may be useful.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $alpha = ⟨ value ⟩$ .
Constraint: $0.0 < alpha < 1.0$ .
NE_WEIGHTS_NOT_POSITIVE: On entry, all elements of w were not positive.
Constraint: $w [i - 1] > 0.0$ , for all $i$ .

7 Accuracy

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

8 Parallelism and Performance

g02abc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02abc 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.

9 Further Comments

Arrays are internally allocated by g02abc. The total size of these arrays is

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

double elements and

5 \times n + 3

Integer elements. All allocated memory is freed before return of g02abc.

10 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})

weighted by

W^{\frac{1}{2}} = diag (100, 20, 20, 20)

with minimum eigenvalue

0.02

g02ab: FL CL CPP AD

NAG CL Interfaceg02abc (corrmat_​nearest_​bounded)

▸▿ 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

NAG CL Interface
g02abc (corrmat_nearest_bounded)