NAG Library Routine Document

d01gyf (md_numth_coeff_prime)


    1  Purpose
    7  Accuracy


d01gyf calculates the optimal coefficients for use by d01gcf and d01gdf, for prime numbers of points.


Fortran Interface
Subroutine d01gyf ( ndim, npts, vk, ifail)
Integer, Intent (In):: ndim, npts
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (Out):: vk(ndim)
C Header Interface
#include nagmk26.h
void  d01gyf_ (const Integer *ndim, const Integer *npts, double vk[], Integer *ifail)


The Korobov (1963) procedure for calculating the optimal coefficients a1,a2,,an for p-point integration over the n-cube 0,1n imposes the constraint that
a1=1  and  ai=ai-1 mod p,  i=1,2,,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
3n01dx101dxni=1n 1-2xi 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 p2) and there is a practical upper limit to the number of points that can be used. Routine d01gzf is computationally more economical in this respect but the associated error is likely to be larger.


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:     npts – IntegerInput
On entry: p, the number of points to be used.
Constraint: npts must be a prime number 5.
3:     vkndim – Real (Kind=nag_wp) arrayOutput
On exit: the n optimal coefficients.
4:     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,npts<5.
On entry,npts is not a prime number.
The precision of the machine is insufficient to perform the computation exactly. Try a smaller value of npts, or use an implementation of 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

d01gyf is not threaded in any implementation.

Further Comments

The time taken is approximately proportional to p2 (see Section 3).


This example calculates the Korobov optimal coefficients where the number of dimensions is 4 and the number of points is 631.

Program Text

Program Text (d01gyfe.f90)

Program Data


Program Results

Program Results (d01gyfe.r)

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