NAG Library Routine Document

1Purpose

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

2Specification

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

3Description

Korobov (1963) gives a procedure for calculating optimal coefficients for $p$-point integration over the $n$-cube ${\left[0,1\right]}^{n}$, when the number of points is
 $p=p1p2$ (1)
where ${p}_{1}$ and ${p}_{2}$ are distinct prime numbers.
The advantage of this procedure is that if ${p}_{1}$ is chosen to be the nearest prime integer to ${p}_{2}^{2}$, then the number of elementary operations required to compute the rule is of the order of ${p}^{4/3}$ 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 $p$.

4References

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

5Arguments

1:     $\mathbf{ndim}$ – IntegerInput
On entry: $n$, the number of dimensions of the integral.
Constraint: ${\mathbf{ndim}}\ge 1$.
2:     $\mathbf{np1}$ – IntegerInput
On entry: the larger prime factor ${p}_{1}$ of the number of points in the integration rule.
Constraint: ${\mathbf{np1}}$ must be a prime number $\text{}\ge 5$.
3:     $\mathbf{np2}$ – IntegerInput
On entry: the smaller prime factor ${p}_{2}$ of the number of points in the integration rule. For maximum efficiency, ${p}_{2}^{2}$ should be close to ${p}_{1}$.
Constraint: ${\mathbf{np2}}$ must be a prime number such that ${\mathbf{np1}}>{\mathbf{np2}}\ge 2$.
4:     $\mathbf{vk}\left({\mathbf{ndim}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the $n$ optimal coefficients.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ndim}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{np1}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np1}}\ge 5$.
On entry, ${\mathbf{np1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np1}}>{\mathbf{np2}}$.
On entry, ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np2}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{np1}}×{\mathbf{np2}}$ exceeds largest machine integer. ${\mathbf{np1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{np1}}=〈\mathit{\text{value}}〉$.
Constraint: np1 must be a prime number.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
Constraint: np2 must be a prime number.
${\mathbf{ifail}}=6$
The machine precision is insufficient to perform the computation exactly. Try reducing np1 or np2: ${\mathbf{np1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

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

8Parallelism and Performance

d01gzf is not threaded in any implementation.

The time taken by d01gzf grows at least as fast as ${\left({p}_{1}{p}_{2}\right)}^{4/3}$. (See Section 3.)

10Example

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 $11$.

10.1Program Text

Program Text (d01gzfe.f90)

None.

10.3Program Results

Program Results (d01gzfe.r)