naginterfaces.library.quad.md_numth_coeff_prime¶
- naginterfaces.library.quad.md_numth_coeff_prime(ndim, npts)[source]¶
md_numth_coeff_prime
calculates the optimal coefficients for use bymd_numth()
andmd_numth_vec()
, for prime numbers of points.For full information please refer to the NAG Library document for d01gy
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d01/d01gyf.html
- Parameters
- ndimint
, the number of dimensions of the integral.
- nptsint
, the number of points to be used.
- Returns
- vkfloat, ndarray, shape
The optimal coefficients.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: must be a prime number.
- Warns
- NagAlgorithmicWarning
- (errno )
The machine precision is insufficient to perform the computation exactly. Try reducing : .
- Notes
The Korobov (1963) procedure for calculating the optimal coefficients for -point integration over the -cube imposes the constraint that
where is a prime number and is an adjustable argument. This argument is computed to minimize the error in the integral
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 is extremely time consuming (the number of elementary operations varying as ) 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