# 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.
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)