NAG FL Interface
d01gzf (md_​numth_​coeff_​2prime)

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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

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 <nag.h>
void  d01gzf_ (const Integer *ndim, const Integer *np1, const Integer *np2, double vk[], Integer *ifail)
The routine may be called by the names d01gzf or nagf_quad_md_numth_coeff_2prime.

3 Description

Korobov (1963) gives a procedure for calculating optimal coefficients for p-point integration over the n-cube [0,1]n, 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 Arguments

1: ndim Integer Input
On entry: n, the number of dimensions of the integral.
Constraint: ndim1.
2: np1 Integer Input
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 Integer Input
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) array Output
On exit: the n optimal coefficients.
5: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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:
ifail=1
On entry, ndim=value.
Constraint: ndim1.
ifail=2
On entry, np1=value.
Constraint: np15.
On entry, np1=value and np2=value.
Constraint: np1>np2.
On entry, np2=value.
Constraint: np22.
ifail=3
On entry, np1×np2 exceeds largest machine integer. np1=value and np2=value.
ifail=4
On entry, np1=value.
Constraint: np1 must be a prime number.
ifail=5
On entry, np2=value.
Constraint: np2 must be a prime number.
ifail=6
The machine precision is insufficient to perform the computation exactly. Try reducing np1 or np2: np1=value and np2=value.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
d01gzf is not threaded in any implementation.

9 Further Comments

The time taken by d01gzf grows at least as fast as (p1p2)4/3. (See Section 3.)

10 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.

10.1 Program Text

Program Text (d01gzfe.f90)

10.2 Program Data

None.

10.3 Program Results

Program Results (d01gzfe.r)