nag_quad_md_numth_coeff_prime (d01gyc) (PDF version)
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NAG Library Manual

# NAG Library Function Documentnag_quad_md_numth_coeff_prime (d01gyc)

## 1  Purpose

nag_quad_md_numth_coeff_prime (d01gyc) calculates the optimal coefficients for use by nag_quad_md_numth_vec (d01gdc), for prime numbers of points.

## 2  Specification

 #include #include
 void nag_quad_md_numth_coeff_prime (Integer ndim, Integer npts, double vk[], NagError *fail)

## 3  Description

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 nag_quad_md_numth_coeff_2prime (d01gzc) is computationally more economical in this respect but the associated error is likely to be larger.

## 4  References

Korobov N M (1963) Number Theoretic Methods in Approximate Analysis Fizmatgiz, Moscow

## 5  Arguments

1:    $\mathbf{ndim}$IntegerInput
On entry: $n$, the number of dimensions of the integral.
Constraint: ${\mathbf{ndim}}\ge 1$.
2:    $\mathbf{npts}$IntegerInput
On entry: $p$, the number of points to be used.
Constraint: ${\mathbf{npts}}$ must be a prime number $\text{}\ge 5$.
3:    $\mathbf{vk}\left[{\mathbf{ndim}}\right]$doubleOutput
On exit: the $n$ optimal coefficients.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error 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 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

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

Not applicable.

## 9  Further Comments

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

## 10  Example

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

### 10.1  Program Text

Program Text (d01gyce.c)

None.

### 10.3  Program Results

Program Results (d01gyce.r)

nag_quad_md_numth_coeff_prime (d01gyc) (PDF version)
d01 Chapter Contents
d01 Chapter Introduction
NAG Library Manual