Numerical evaluation of multiple integrals. II

Abstract

Introduction. Several specific methods for numerical evaluation of integrals over higher dimensional regions have been proposed. James Clerk-Maxwell [_1 j proposed the formulas for the rectangle and the rectangular parallelopipedon in 1877. Appell, Burnside, Ionescue, and Mineur have developed special formulas for planar regions. Tyler [2] recently gave formulas for rectangles, parabolic regions, cubes, and for the hypercubes. Others have developed formulas pri-marily for rectangular regions based on the formulas for the line. Although the original draft of this paper antedates their results, Hammer, Marlowe and Stroud [3] have given an inductive method for numerical evaluation of integrals precise for ¿th degree polynomials over the «-simplex. They also show how to obtain formulas for cones based on regions for which integration formulas are given. Certain affinely symmetrical formulas for triangle and tetrahedron are given in their paper. In a sequel Hammer and Stroud [4] gave affinely sym-metrical formulas precise for the quadratic and cubic polynomials over the re-simplex. W. H. Peirce £5] in his doctoral dissertation has given numerical integration formulas over the circular annulus precise for arbitrarily high degree polynomials. Despite the variety of results and their particular interest, the theory of numerical integration in higher dimensions is in a very crude state of development in comparison with numerical integration of functions of one variable. For this reason we give here a few theorems which are actually quite simple but which evidently have escaped the awareness of most research workers. These theorems form a partial basis for development of classical type integration formulas. The sole numerical integration device with claims to somewhat universal applicability is that called the Monte Carlo method which has been extended and popularized by S. M. Ulam and John von Neumann. Granted the presence of a suitable sequence of "random" digits or the means of generating such, the Monte Carlo methods can be applied to a large variety of problems by using high-speed digital computers. It has been stated as an advantage of Monte Carlo methods that the number of evaluation points for, say, 5 percent error does not mount as rapidly for higher dimensions as it does for classical formulas. We shall show (as Tyler, and Hammer and Stroud have shown in other cases) that the evaluation points for classical type formulas can conceivably be kept down very well. However, the problem of determining points and weights in these circum-stances is a problem in the theory of equations which is difficult inherently and also because of the great variety of functions and regions of conceivable interest. In this direction the Monte Carlo method is less sensitive to the deviations among regions. A Basic Theorem for Numerical Integration. In this section we state a theorem which greatly extends the usefulness of particular integration formulas when they are available. We limit the discussion to integration formulas of the form