QUADRATURE METHODS FOR FUNCTIONS OF MORE THAN ONE VARIABLE

Here f(xl, . . . , z,) is a real-valued function of n red variables (n 2 2) ; R is a region in n-dimensional, real, Euclidean space; the A , are constants independent of f ; and the v, = ( vll , . . . , 'vim) are points in the space. I shall summarize some of the most important results and not try to survey the literature completely. I shall also point out some of the most interesting problems that remain to be investigated. The comparatively recent development of high-speed computing machines has made it possible to solve more difficult problems in higher dimensions and is probably largely responsible for the recent interest in approximate integration for functions of more than one variable. The Monte Carlo methods of integration have been used widely with these computers. With the Monte Carlo methods-for example, the method of choosing the points in a random manner in R and taking the Ai all equal to (vol. R ) / (number of points)-one usually expects to use a large number of points, perhaps several thousand, to obtain a close approximation. The approximation method in which I shall be chiefly interested here is one in which we choose the A i and the vi so that the approximation is exact whenever f is a polynomial of degree Slc for various positive integers 1c (that is, whenever