NAG Library Routine Document

d01gzf (md_numth_coeff_2prime)


    1  Purpose
    7  Accuracy


d01gzf calculates the optimal coefficients for use by d01gcf and d01gdf, when the number of points is the product of two primes.


Fortran Interface
Subroutine d01gzf ( ndim, np1, np2, vk, ifail)
Integer, Intent (In):: ndim, np1, np2
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (Out):: vk(ndim)
C Header Interface
#include nagmk26.h
void  d01gzf_ (const Integer *ndim, const Integer *np1, const Integer *np2, double vk[], Integer *ifail)


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.


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


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:     vkndim – 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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

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.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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

Parallelism and Performance

d01gzf is not threaded in any implementation.

Further Comments

The time taken by d01gzf grows at least as fast as p1p24/3. (See Section 3.)


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.

Program Text

Program Text (d01gzfe.f90)

Program Data


Program Results

Program Results (d01gzfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017