NAG Library Function Document

nag_quad_md_numth_coeff_prime (d01gyc)

+− 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

1 Purpose

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ACCURACY: The machine precision is insufficient to perform the computation exactly. Try reducing npts: $npts = ⟨value⟩$ .
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.