nag_quad_md_numth_coeff_2prime (d01gz) calculates the optimal coefficients for use by nag_quad_md_numth (d01gc) and nag_quad_md_numth_vec (d01gd), when the number of points is the product of two primes.

Syntax

[vk, ifail] = d01gz(ndim, np1, np2)

[vk, ifail] = nag_quad_md_numth_coeff_2prime(ndim, np1, np2)

Description

Korobov (1963) gives a procedure for calculating optimal coefficients for

p

-point integration over the

n

-cube

{[0, 1]}^{n}

, when the number of points is

p = p_{1} p_{2}

(1)

where

p_{1}

and

p_{2}

are distinct prime numbers.

The advantage of this procedure is that if

p_{1}

is chosen to be the nearest prime integer to

p_{2}^{2}

, then the number of elementary operations required to compute the rule is of the order of

p^{4 / 3}

which grows less rapidly than the number of operations required by nag_quad_md_numth_coeff_prime (d01gy). The associated error is likely to be larger although it may be the only practical alternative for high values of

p

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: $np1$ – int64int32nag_int scalar: The larger prime factor $p_{1}$ of the number of points in the integration rule.

Constraint: $np1$ must be a prime number $\geq 5$ .
3: $np2$ – int64int32nag_int scalar: The smaller prime factor $p_{2}$ of the number of points in the integration rule. For maximum efficiency, $p_{2}^{2}$ should be close to $p_{1}$ .

Constraint: $np2$ must be a prime number such that $np1 > np2 \geq 2$ .

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,	$np1 < 5$ ,
or	$np2 < 2$ ,
or	$np1 \leq np2$ .

$ifail = 3$: The value $np1 \times np2$ exceeds the largest integer representable on the machine, and hence the optimal coefficients could not be used in a valid call of nag_quad_md_numth (d01gc) or nag_quad_md_numth_vec (d01gd).

$ifail = 4$

On entry,

np1 is not a prime number.

$ifail = 5$

On entry,

np2 is not a prime number.

W $ifail = 6$: The precision of the machine is insufficient to perform the computation exactly. Try smaller values of np1 or np2, or use an implementation with 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 by nag_quad_md_numth_coeff_2prime (d01gz) grows at least as fast as

{(p_{1} p_{2})}^{4 / 3}

. (See Description.)

Example

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

4

and the number of points is the product of the two prime numbers,

89

and

11

Open in the MATLAB editor: d01gz_example

function d01gz_example


fprintf('d01gz example results\n\n');

ndim = int64(4);
np1 = int64(89);
np2 = int64(11);

[vk, ifail] = d01gz(ndim, np1, np2);

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

d01gz example results

Optimal coefficients:     1   102   614   951

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

Chapter Contents

Chapter Introduction

NAG Toolbox