Numerical iterated integration based on the double exponential transformation

Abstract

A formula for numerical evaluation of two dimensional iterated integrals of the formI = ∫ ^b _a dx ∫ ^q(x) _a′ f(x,y)dy whereq(a) =a’,q(b) =b’ (a <x <b) is derived by means of the sinc approximation based on the double exponential transformation. The integrand ƒ(x,y) is assumed to be analytic as a function of x on a < x < b and also of y on a’ < y < b’, and q(x) is assumed to be an analytic function of x on a < x < b. The order of the error of the formula derived is O (exp(−βN/ log(γN))) as a function of$N = \sqrt {n_{total} } /2$ wheren _total is the total number of function evaluations. Numerical examples also proves high efficiency of the formula. When the integrand is of a product type, i.e. ƒ(x,y) = X(x)Y(y), we can obtain an approximate value of I by evaluating only 2 × (2N + 1) values (X(x_j),−N ≤ j ≤ N), (Y(yk), −N ≤ k ≤ N), and hence the number of function evaluations is reduced to O(4N).