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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_quad_md_numth_coeff_2prime (d01gz)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


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.


[vk, ifail] = d01gz(ndim, np1, np2)
[vk, ifail] = nag_quad_md_numth_coeff_2prime(ndim, np1, np2)


Korobov (1963) gives a procedure for calculating optimal coefficients for p-point integration over the n-cube 0,1n, when the number of points is
p=p1p2 (1)
where p1 and p2 are distinct prime numbers.
The advantage of this procedure is that if p1 is chosen to be the nearest prime integer to p22, then the number of elementary operations required to compute the rule is of the order of p4/3 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.


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


Compulsory Input Parameters

1:     ndim int64int32nag_int scalar
n, the number of dimensions of the integral.
Constraint: ndim1.
2:     np1 int64int32nag_int scalar
The larger prime factor p1 of the number of points in the integration rule.
Constraint: np1 must be a prime number 5.
3:     np2 int64int32nag_int scalar
The smaller prime factor p2 of the number of points in the integration rule. For maximum efficiency, p22 should be close to p1.
Constraint: np2 must be a prime number such that np1>np22.

Optional Input Parameters


Output Parameters

1:     vkndim – 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.

On entry,ndim<1.
On entry,np1<5,
The value np1×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).
On entry,np1 is not a prime number.
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.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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 p1p24/3. (See Description.)


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.
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:');

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

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