nag_quad_md_numth_coeff_2prime (d01gzc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_quad_md_numth_coeff_2prime (d01gzc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_quad_md_numth_coeff_2prime (d01gzc) calculates the optimal coefficients for use by nag_quad_md_numth_vec (d01gdc), when the number of points is the product of two primes.

2  Specification

#include <nag.h>
#include <nagd01.h>
void  nag_quad_md_numth_coeff_2prime (Integer ndim, Integer np1, Integer np2, double vk[], NagError *fail)

3  Description

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 (d01gyc). The associated error is likely to be larger although it may be the only practical alternative for high values of p.

4  References

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

5  Arguments

1:     ndimIntegerInput
On entry: n, the number of dimensions of the integral.
Constraint: ndim1.
2:     np1IntegerInput
On entry: the larger prime factor p1 of the number of points in the integration rule.
Constraint: np1 must be a prime number 5.
3:     np2IntegerInput
On entry: 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.
4:     vk[ndim]doubleOutput
On exit: the n optimal coefficients.
5:     failNagError *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 np1 or np2: np1=value and np2=value.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, ndim=value.
Constraint: ndim1.
On entry, np1=value.
Constraint: np1 must be a prime number.
On entry, np1=value.
Constraint: np15.
On entry, np2=value.
Constraint: np2 must be a prime number.
On entry, np2=value.
Constraint: np22.
NE_INT_2
On entry, np1×np2 exceeds largest machine integer. np1=value and np2=value.
On entry, np1=value and np2=value.
Constraint: np1>np2.
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.

7  Accuracy

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

8  Further Comments

The time taken by nag_quad_md_numth_coeff_2prime (d01gzc) grows at least as fast as p1p24/3. (See Section 3.)

9  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.

9.1  Program Text

Program Text (d01gzce.c)

9.2  Program Data

Program Data (d01gzce.d)

9.3  Program Results

Program Results (d01gzce.r)


nag_quad_md_numth_coeff_2prime (d01gzc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012