nag_quad_md_numth_coeff_prime (d01gyc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_quad_md_numth_coeff_prime (d01gyc)


    1  Purpose
    7  Accuracy

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 a1,a2,,an for p-point integration over the n-cube 0,1n imposes the constraint that
a1=1  and  ai=ai-1 mod p,  i=1,2,,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
3n01dx101dxni=1n 1-2xi 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 p2) 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: ndim1.
2:     npts IntegerInput
On entry: p, the number of points to be used.
Constraint: npts must be a prime number 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

The machine precision is insufficient to perform the computation exactly. Try reducing npts: npts=value.
Dynamic memory allocation failed.
See Section in the Essential Introduction for further information.
On entry, argument value had an illegal value.
On entry, ndim=value.
Constraint: ndim1.
On entry, npts=value.
Constraint: npts must be a prime number.
On entry, npts=value.
Constraint: npts5.
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

7  Accuracy

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

8  Parallelism and Performance

Not applicable.

9  Further Comments

The time taken is approximately proportional to p2 (see Section 3).

10  Example

This example calculates the Korobov optimal coefficients where the number of dimensions is 4 and the number of points is 631.

10.1  Program Text

Program Text (d01gyce.c)

10.2  Program Data


10.3  Program Results

Program Results (d01gyce.r)

nag_quad_md_numth_coeff_prime (d01gyc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015