抄録
A formula for numerical evaluation of iterated integrals of the form I = ∫^b_a dx ∫^<q(x)>_c f(x,y)dy where q(a) = c, q(b) = d (a < x < b) is derived by means of the double exponential transformation followed by the sine approximation. The integrand f(x,y) is assumed to be analytic as a function of x in a < x < b and also of y in c < y < d, and q(x) is assumed to be an analytic function of x in a < x < b. The error of the formula derived is O (exp(-βN/log(γN))) as a function of N = (√<n_<total>> - 1)/2 where n_<total> is the total number of function evaluations. When the integrand has a split form f(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(y_k), -N ≤ k ≤ N), and hence the number of function evaluations is reduced to O(N). Numerical example also indicate high efficiency of the formula.
