二重指数関数型変換による累次積分の計算

概要

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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.
日本応用数理学会の論文
2003-12-25