NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01gy: FL CL AD

NAG CL Interface
d01gyc (md_numth_coeff_prime)

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NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01gy: FL CL AD

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2019

1 Purpose

d01gyc calculates the optimal coefficients for use by d01gdc, for prime numbers of points.

2 Specification

#include <nag.h>

void	d01gyc (Integer ndim, Integer npts, double vk[], NagError *fail)

The function may be called by the names: d01gyc or nag_quad_md_numth_coeff_prime.

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 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: $ndim$ – Integer Input: On entry: $n$ , the number of dimensions of the integral.

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

Constraint: $npts$ must be a prime number $\geq 5$ .
3: $vk [ndim]$ – double Output: On exit: the $n$ optimal coefficients.
4: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

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

8 Parallelism and Performance

d01gyc is not threaded in any implementation.

9 Further Comments

The time taken is approximately proportional to

p^{2}

(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

None.

10.3 Program Results

Program Results (d01gyce.r)

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01gy: FL CL AD

© The Numerical Algorithms Group Ltd. 2019