D01 Chapter Contents
D01 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentD01GYF

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.

1  Purpose

D01GYF calculates the optimal coefficients for use by D01GCF and D01GDF, for prime numbers of points.

2  Specification

 SUBROUTINE D01GYF ( NDIM, NPTS, VK, IFAIL)
 INTEGER NDIM, NPTS, IFAIL REAL (KIND=nag_wp) VK(NDIM)

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 ${\left[0,1\right]}^{n}$ imposes the constraint that
 (1)
where $p$ is a prime number and $a$ is an adjustable parameter. This parameter is computed to minimize the error in the integral
 $3n∫01dx1⋯∫01dxn∏i=1n 1-2xi 2,$ (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 ${p}^{2}$) 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.

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: ${\mathbf{NDIM}}\ge 1$.
2:     NPTS – INTEGERInput
On entry: $p$, the number of points to be used.
Constraint: ${\mathbf{NPTS}}$ must be a prime number $\text{}\ge 5$.
3:     VK(NDIM) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ optimal coefficients.
4:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{NDIM}}<1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{NPTS}}<5$.
${\mathbf{IFAIL}}=3$
 On entry, NPTS is not a prime number.
${\mathbf{IFAIL}}=4$
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.

7  Accuracy

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

The time taken is approximately proportional to ${p}^{2}$ (see Section 3).

9  Example

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

9.1  Program Text

Program Text (d01gyfe.f90)

None.

9.3  Program Results

Program Results (d01gyfe.r)