nag_quad_md_numth_coeff_prime (d01gy) calculates the optimal coefficients for use by nag_quad_md_numth (d01gc) and nag_quad_md_numth_vec (d01gd), for prime numbers of points.

Syntax

[vk, ifail] = d01gy(ndim, npts)

[vk, ifail] = nag_quad_md_numth_coeff_prime(ndim, npts)

Description

The Korobov (1963) procedure for calculating the optimal coefficients

a_{1}, a_{2}, \dots, a_{n}

for

p

-point integration over the

n

-cube

{[0, 1]}^{n}

imposes the constraint that

a_{1} = 1 and a_{i} = a^{i - 1} (\mod p), i = 1, 2, \dots, n

(1)

where

p

is a prime number and

a

is an adjustable argument. This argument is computed to minimize the error in the integral

3^{n} \int_{0}^{1} d x_{1} \dots \int_{0}^{1} d x_{n} \prod_{i = 1}^{n} {(1 - 2 x_{i})}^{2},

(2)

when computed using the number theoretic rule, and the resulting coefficients can be shown to fit the Korobov definition of optimality.

The computation for large values of

p

is extremely time consuming (the number of elementary operations varying as

p^{2}

) and there is a practical upper limit to the number of points that can be used. Function nag_quad_md_numth_coeff_2prime (d01gz) is computationally more economical in this respect but the associated error is likely to be larger.

References

Korobov N M (1963) Number Theoretic Methods in Approximate Analysis Fizmatgiz, Moscow

Parameters

Compulsory Input Parameters

1: $ndim$ – int64int32nag_int scalar: $n$ , the number of dimensions of the integral.

Constraint: $ndim \geq 1$ .
2: $npts$ – int64int32nag_int scalar: $p$ , the number of points to be used.

Constraint: $npts$ must be a prime number $\geq 5$ .

Optional Input Parameters

None.

Output Parameters

1: $vk (ndim)$ – double array: The $n$ optimal coefficients.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,

ndim < 1

$ifail = 2$

On entry,

npts < 5

$ifail = 3$

On entry,

npts is not a prime number.

W $ifail = 4$: The precision of the machine is insufficient to perform the computation exactly. Try a smaller value of npts, or use an implementation of higher precision.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The optimal coefficients are returned as exact integers (though stored in a double array).

Further Comments

The time taken is approximately proportional to

p^{2}

(see Description).

Example

This example calculates the Korobov optimal coefficients where the number of dimensions is

4

and the number of points is

631

Open in the MATLAB editor: d01gy_example

function d01gy_example


fprintf('d01gy example results\n\n');

ndim = int64(4);
npts = int64(631);

[vk, ifail] = d01gy(ndim, npts);

fprintf('Optimal coefficients:');
fprintf('%6d',vk);
fprintf('\n');

d01gy example results

Optimal coefficients:     1   198    82   461

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox