D01GZF (PDF version)
D01 Chapter Contents
D01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

D01GZF calculates the optimal coefficients for use by D01GCF and D01GDF, when the number of points is the product of two primes.

2  Specification

REAL (KIND=nag_wp)  VK(NDIM)

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 D01GYF. 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  Parameters

1:     NDIM – INTEGERInput
On entry: n, the number of dimensions of the integral.
Constraint: NDIM1.
2:     NP1 – INTEGERInput
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:     NP2 – INTEGERInput
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) – REAL (KIND=nag_wp) arrayOutput
On exit: the n optimal coefficients.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry,NDIM<1.
On entry,NP1<5,
The value NP1×NP2 exceeds the largest integer representable on the machine, and hence the optimal coefficients could not be used in a valid call of D01GCF or D01GDF.
On entry,NP1 is not a prime number.
On entry,NP2 is not a prime number.
The precision of the machine is insufficient to perform the computation exactly. Try smaller values of NP1 or NP2, or use an implementation with higher precision.

7  Accuracy

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

8  Further Comments

The time taken by D01GZF 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 (d01gzfe.f90)

9.2  Program Data


9.3  Program Results

Program Results (d01gzfe.r)

D01GZF (PDF version)
D01 Chapter Contents
D01 Chapter Introduction
NAG Library Manual

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