Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals

Abstract

A new method, of very general applicability and very easily programmed for an electronic computer, is proposed for the numerical integration of functions of many independent variables. This new method renders obsolete, in most applications, the commonly used Monte Carlo procedure and the more recent, original method of Haselgrove. In the new scheme the sample points are distributed systematically rather than at random and the ensemble of points forms a unique, closed, symmetrical pattern, which effectively fills the space of the multidimensional integration. The paper contains an extensive statistical‐analytic treatment of the error characteristics of the new method, one that enables advance quantitative estimation of upper limits of error in the integration of various types of functions. For continuous functions with bounded first derivatives, the error is shown ultimately to disappear at least as rapidly as the inverse square of the number of sample points; moreover, for runs of practical length the error limits with the new scheme are smaller—by a factor ranging from 2 to perhaps 10⁴ or more—than those of any previous general procedure. The method employs certain rational constants which govern the arrangement of sample points. Tables of such constants, suitably optimized, which will permit the integration of functions with up to 12 independent variables, are provided along with a discussion of a method by which such constants may be obtained.