naginterfaces.library.quad.md_numth_coeff_2prime¶

naginterfaces.library.quad.md_numth_coeff_2prime(ndim, np1, np2)[source]¶

md_numth_coeff_2prime calculates the optimal coefficients for use by md_numth() and md_numth_vec(), when the number of points is the product of two primes.

For full information please refer to the NAG Library document for d01gz

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01gzf.html

Parameters

ndimint: $n$ , the number of dimensions of the integral.
np1int: The larger prime factor $p_{1}$ of the number of points in the integration rule.
np2int: The smaller prime factor $p_{2}$ of the number of points in the integration rule. For maximum efficiency, $p_{2}^{2}$ should be close to $p_{1}$ .

Returns

vkfloat, ndarray, shape $(n d i m)$: The $n$ optimal coefficients.

Raises

NagValueError

(errno $1$ )

On entry, $n d i m = ⟨ v a l u e ⟩$ .

Constraint: $n d i m \geq 1$ .

(errno $2$ )

On entry, $n p 1 = ⟨ v a l u e ⟩$ and $n p 2 = ⟨ v a l u e ⟩$ .

Constraint: $n p 1 > n p 2$ .

(errno $2$ )

On entry, $n p 2 = ⟨ v a l u e ⟩$ .

Constraint: $n p 2 \geq 2$ .

(errno $2$ )

On entry, $n p 1 = ⟨ v a l u e ⟩$ .

Constraint: $n p 1 \geq 5$ .

(errno $3$ )

On entry, $n p 1 \times n p 2$ exceeds largest machine integer. $n p 1 = ⟨ v a l u e ⟩$ and $n p 2 = ⟨ v a l u e ⟩$ .

(errno $4$ )

On entry, $n p 1 = ⟨ v a l u e ⟩$ .

Constraint: $n p 1$ must be a prime number.

(errno $5$ )

On entry, $n p 2 = ⟨ v a l u e ⟩$ .

Constraint: $n p 2$ must be a prime number.

Warns

NagAlgorithmicWarning

(errno $6$ ): The machine precision is insufficient to perform the computation exactly. Try reducing $n p 1$ or $n p 2$ : $n p 1 = ⟨ v a l u e ⟩$ and $n p 2 = ⟨ v a l u e ⟩$ .

Notes

Korobov (1963) gives a procedure for calculating optimal coefficients for $p$ -point integration over the $n$ -cube ${[0, 1]}^{n}$ , when the number of points is

p = p_{1} p_{2}

where $p_{1}$ and $p_{2}$ are distinct prime numbers.

The advantage of this procedure is that if $p_{1}$ is chosen to be the nearest prime integer to $p_{2}^{2}$ , then the number of elementary operations required to compute the rule is of the order of $p^{4 / 3}$ which grows less rapidly than the number of operations required by md_numth_coeff_prime(). The associated error is likely to be larger although it may be the only practical alternative for high values of $p$ .

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

NAG and Python

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naginterfaces.library.quad.md_numth_coeff_2prime¶

naginterfaces.library.quad.md_​numth_​coeff_​2prime¶

naginterfaces.library.quad.md_numth_coeff_2prime¶