# NAG CL Interfaced01gyc (md_​numth_​coeff_​prime)

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## 1Purpose

d01gyc calculates the optimal coefficients for use by d01gdc, for prime numbers of points.

## 2Specification

 #include
 void d01gyc (Integer ndim, Integer npts, double vk[], NagError *fail)
The function may be called by the names: d01gyc or nag_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. Function d01gzc 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: $\mathbf{vk}\left[{\mathbf{ndim}}\right]$double Output
On exit: the $n$ optimal coefficients.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ACCURACY
The machine precision is insufficient to perform the computation exactly. Try reducing npts: ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{ndim}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ndim}}\ge 1$.
On entry, ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
Constraint: npts must be a prime number.
On entry, ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{npts}}\ge 5$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

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

## 8Parallelism and Performance

d01gyc 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 (d01gyce.c)

### 10.2Program Data

Program Data (d01gyce.d)

### 10.3Program Results

Program Results (d01gyce.r)