NAG C Library Function Document

nag_quad_md_numth_coeff_prime (d01gyc) calculates the optimal coefficients for use by nag_quad_md_numth_vec (d01gdc), for prime numbers of points.

2

Specification

#include <nag.h>

#include <nagd01.h>

void	nag_quad_md_numth_coeff_prime (Integer ndim, Integer npts, double vk[], NagError *fail)

3

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 (d01gzc) is computationally more economical in this respect but the associated error is likely to be larger.

4

References

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

5

Arguments

1: $ndim$ – IntegerInput: On entry: $n$ , the number of dimensions of the integral.

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

Constraint: $npts$ must be a prime number $\geq 5$ .
3: $vk [ndim]$ – doubleOutput: On exit: the $n$ optimal coefficients.
4: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ACCURACY: The machine precision is insufficient to perform the computation exactly. Try reducing npts: $npts = 〈value〉$ .
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_INT: On entry, $ndim = 〈value〉$ .
Constraint: $ndim \geq 1$ .

On entry, $npts = 〈value〉$ .
Constraint: npts must be a prime number.

On entry, $npts = 〈value〉$ .
Constraint: $npts \geq 5$ .
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 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.