naginterfaces.library.quad.md_​numth_​coeff_​prime

naginterfaces.library.quad.md_numth_coeff_prime(ndim, npts)

md_numth_coeff_prime calculates the optimal coefficients for use by md_numth() and md_numth_vec(), for prime numbers of points.

For full information please refer to the NAG Library document for d01gy

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/d01/d01gyf.html

Parameters

ndimint

$n$, the number of dimensions of the integral.

nptsint

$p$, the number of points to be used.

Returns

vkfloat, ndarray, shape $\left(ndim\right)$

The $n$ optimal coefficients.

Raises

NagValueError

(errno $1$)

On entry, $ndim=⟨value⟩$.

Constraint: $ndim\ge 1$.

(errno $2$)

On entry, $npts=⟨value⟩$.

Constraint: $npts\ge 5$.

(errno $3$)

On entry, $npts=⟨value⟩$.

Constraint: $npts$ must be a prime number.

Warns

NagAlgorithmicWarning

(errno $4$)

The machine precision is insufficient to perform the computation exactly. Try reducing $npts$: $npts=⟨value⟩$.

Notes

The Korobov (1963) procedure for calculating the optimal coefficients $a_1,a_2,\dots ,a_n$ for $p$-point integration over the $n$-cube ${[0,1]}^n$ imposes the constraint that

$$a_1=1\text{and}{a}_i=a^{i-1}\left(\text{mod}p\right)\text{,}i=1,2,\dots ,n$$

where $p$ is a prime number and $a$ is an adjustable argument. This argument is computed to minimize the error in the integral

$$3^n{\int }_0^{1}d{x}_1\cdots {\int }_0^{1}d{x}_n\prod _{i=1}^n{(1-2x_i)}^2\text{,}$$

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 md_numth_coeff_2prime() is computationally more economical in this respect but the associated error is likely to be larger.

References

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