NAG Library Function Document

nag_quad_md_numth_coeff_2prime (d01gzc)

+− 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_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, 1]}^{n}

, when the number of points is

p = p_{1} p_{2}

(1)

where

p_{1}

and

p_{2}

are distinct prime numbers.

The advantage of this procedure is that if

p_{1}

is chosen to be the nearest prime integer to

p_{2}^{2}

, then the number of elementary operations required to compute the rule is of the order of

p^{4 / 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: ndim – IntegerInput: On entry: $n$ , the number of dimensions of the integral.
Constraint: $ndim \geq 1$ .
2: np1 – IntegerInput: On entry: the larger prime factor $p_{1}$ of the number of points in the integration rule.
Constraint: $np1$ must be a prime number $\geq 5$ .
3: np2 – IntegerInput: On entry: the smaller prime factor $p_{2}$ of the number of points in the integration rule. For maximum efficiency, $p_{2}^{2}$ should be close to $p_{1}$ .
Constraint: $np2$ must be a prime number such that $np1 > np2 \geq 2$ .
4: vk[ndim] – doubleOutput: On exit: the $n$ optimal coefficients.
5: 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 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: $ndim \geq 1$ .
On entry, $np1 = ⟨value⟩$ .
Constraint: np1 must be a prime number.
On entry, $np1 = ⟨value⟩$ .
Constraint: $np1 \geq 5$ .
On entry, $np2 = ⟨value⟩$ .
Constraint: np2 must be a prime number.
On entry, $np2 = ⟨value⟩$ .
Constraint: $np2 \geq 2$ .
NE_INT_2: On entry, $np1 \times np2$ 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 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_quad_md_numth_coeff_2prime (d01gzc) grows at least as fast as

{(p_{1} p_{2})}^{4 / 3}

. (See Section 3.)

10 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

10.1 Program Text

Program Text (d01gzce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (d01gzce.r)

nag_quad_md_numth_coeff_2prime (d01gzc) (PDF version)

d01 Chapter Contents

d01 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_quad_md_numth_coeff_2prime (d01gzc)