# NAG FL Interfaced01gzf (md_​numth_​coeff_​2prime)

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## 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 <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.

## 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: $\mathbf{ndim}$Integer Input
On entry: $n$, the number of dimensions of the integral.
Constraint: ${\mathbf{ndim}}\ge 1$.
2: $\mathbf{np1}$Integer Input
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: $\mathbf{np2}$Integer Input
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) array Output
On exit: the $n$ optimal coefficients.
5: $\mathbf{ifail}$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 $-\mathbf{1}$ 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).

## 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 ${\mathbf{np2}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)