naginterfaces.library.quad.md_numth(f, region, npts, vk, nrand, itrans=0, data=None)[source]

md_numth calculates an approximation to a definite integral in up to dimensions, using the Korobov–Conroy number theoretic method.

For full information please refer to the NAG Library document for d01gc

fcallable retval = f(x, data=None)

must return the value of the integrand at a given point.

xfloat, ndarray, shape

The coordinates of the point at which the integrand must be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.


The value of evaluated at .

regioncallable (c, d) = region(x, j, data=None)

must evaluate the limits of integration in any dimension.

xfloat, ndarray, shape

contain the current values of the first variables, which may be used if necessary in calculating and .


The index for which the limits of the range of integration are required.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.


The lower limit of the range of .


The upper limit of the range of .


The Korobov rule to be used. There are two alternatives depending on the value of .

  1. .

    In this case one of six preset rules is chosen using , , , , or points depending on the respective value of being , , , , or .

  2. .

    is the number of actual points to be used with corresponding optimal coefficients supplied in the array .

vkfloat, array-like, shape

If , must contain the optimal coefficients (which may be calculated using md_numth_coeff_prime() or md_numth_coeff_2prime()).

If , need not be set.


The number of random samples to be generated in the error estimation (generally a small value, say to , is sufficient). The total number of integrand evaluations will be .

itransint, optional

Indicates whether the periodising transformation is to be used.

The transformation is to be used.

The transformation is to be suppressed (to cover cases where the integrand may already be periodic or where you want to specify a particular transformation in the definition of ).

dataarbitrary, optional

User-communication data for callback functions.

vkfloat, ndarray, shape

If , is unchanged.

If , contains the optimal coefficients used by the preset rule.


The approximation to the integral .


The standard error as computed from sample values. If , contains zero.

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .


No equivalent traditional C interface for this routine exists in the NAG Library.

md_numth calculates an approximation to the integral

using the Korobov–Conroy number theoretic method (see Korobov (1957), Korobov (1963) and Conroy (1967)). The region of integration defined in (1) is such that generally and may be functions of , for , with and constants. The integral is first of all transformed to an integral over the -cube by the change of variables

The method then uses as its basis the number theoretic formula for the -cube, :

where denotes the fractional part of , are the so-called optimal coefficients, is the error, and is a prime integer. (It is strictly only necessary that be relatively prime to all and is in fact chosen to be even for some cases in Conroy (1967).) The method makes use of properties of the Fourier expansion of which is assumed to have some degree of periodicity. Depending on the choice of the contributions from certain groups of Fourier coefficients are eliminated from the error, . Korobov shows that can be chosen so that the error satisfies

where and are real numbers depending on the convergence rate of the Fourier series, is a constant depending on , and is a constant depending on and . There are a number of procedures for calculating these optimal coefficients. Korobov imposes the constraint that

and gives a procedure for calculating the argument, , to satisfy the optimal conditions.

In this function the periodisation is achieved by the simple transformation

More sophisticated periodisation procedures are available but in practice the degree of periodisation does not appear to be a critical requirement of the method.

An easily calculable error estimate is not available apart from repetition with an increasing sequence of values of which can yield erratic results. The difficulties have been studied by Cranley and Patterson (1976) who have proposed a Monte Carlo error estimate arising from converting (2) into a stochastic integration rule by the inclusion of a random origin shift which leaves the form of the error (3) unchanged; i.e., in the formula (2), is replaced by , for , where each , is uniformly distributed over . Computing the integral for each of a sequence of random vectors allows a ‘standard error’ to be estimated.

This function provides built-in sets of optimal coefficients, corresponding to six different values of . Alternatively, the optimal coefficients may be supplied by you. Functions md_numth_coeff_prime() and md_numth_coeff_2prime() compute the optimal coefficients for the cases where is a prime number or is a product of two primes, respectively.


Conroy, H, 1967, Molecular Shroedinger equation VIII. A new method for evaluting multi-dimensional integrals, J. Chem. Phys. (47), 5307–5318

Cranley, R and Patterson, T N L, 1976, Randomisation of number theoretic methods for mulitple integration, SIAM J. Numer. Anal. (13), 904–914

Korobov, N M, 1957, The approximate calculation of multiple integrals using number theoretic methods, Dokl. Acad. Nauk SSSR (115), 1062–1065

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