# NAG FL Interfaced01gyf (md_​numth_​coeff_​prime)

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## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine d01gyf ( ndim, npts, vk,
 Integer, Intent (In) :: ndim, npts Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Out) :: vk(ndim)
#include <nag.h>
 void d01gyf_ (const Integer *ndim, const Integer *npts, double vk[], Integer *ifail)
The routine may be called by the names d01gyf or nagf_quad_md_numth_coeff_prime.

## 3Description

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 argument. This argument 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.

## 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: $\mathbf{npts}$Integer Input
On entry: $p$, the number of points to be used.
Constraint: ${\mathbf{npts}}$ must be a prime number $\text{}\ge 5$.
3: $\mathbf{vk}\left({\mathbf{ndim}}\right)$Real (Kind=nag_wp) array Output
On exit: the $n$ optimal coefficients.
4: $\mathbf{ifail}$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, ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{npts}}\ge 5$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
Constraint: npts must be a prime number.
${\mathbf{ifail}}=4$
The machine precision is insufficient to perform the computation exactly. Try reducing npts: ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
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

d01gyf is not threaded in any implementation.

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

## 10Example

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

### 10.1Program Text

Program Text (d01gyfe.f90)

None.

### 10.3Program Results

Program Results (d01gyfe.r)