NAG Library Function Document

1Purpose

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.

2Specification

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

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 nag_quad_md_numth_coeff_2prime (d01gzc) 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:    $\mathbf{ndim}$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 3.7 in How to Use the NAG Library and its Documentation).

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 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

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

8Parallelism and Performance

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)

None.

10.3Program Results

Program Results (d01gyce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017