Numerical calculation of integrals with strongly oscillating integrand
Numerical calculation of integrals with strongly oscillating integrand
复制标题
强振荡被积函数积分的数值计算
DOI：
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发表时间：
1966
期刊：
影响因子：
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通讯作者：
H. Linde
中科院分区：
文献类型：
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作者：
A. I. Vooren;H. Linde
本文给出了求Njf（x）eiwxdx的一种方法，其中wN = p-2 r，p为整数.其思想是借助于多项式来逼近f（x）而不是整个被积函数. Romberg-Stiefel算法已被扩展到这种情况。新的方法是补充通常的Romberg-Stiefel算法在这个意义上，它是更有利的较大值的a。还包括余数项的表达式。如果f（x）至多为7次，则真实的部分的结果是精确的，如果f（x）至多为8次，则虚部的结果是精确的。1.导论.传统的数值积分方法一般不太适合计算9 P 00 f（x）cos xx dx和ff（x）sin xx dx形式的积分，如果X很大。由于被积函数的振荡特性，它的近似多项式的援助需要大量的点，被积函数必须评估。本文给出了一种方法，即用函数f（x）代替整个被积函数用多项式逼近。此外，该方法的纯数值部分仅限于N rN的评估
In this paper a method is presented for evaluating N j f(x)eiwx dx where wN = p- 2r, p integer. The idea is to approximate f(x) instead of the whole integrand by aid of poly- nomials. The Romberg-Stiefel algorithm has been extended to this case. The new method is complementary to the usual Romberg-Stiefel algorithm in the sense that it is more advantageous for larger values of a. An expression for the remainder term is also included. Results for the real part are exact if f(x) is of at most 7th degree and for the imaginary part if f(x) is of at most 8th degree. 1. Introduction. The conventional methods of numerical integration are generally less suitable for the computation of integrals of the form 9P 00 f(x) cos xx dx and ff(x) sin xx dx, if X is large. Due to the oscillatory character of the integrand, its approximation by aid of polynomials requires a large number of points where the integrand must be evaluated. In the present paper a method is given where instead of the whole integrand only the function f(x) is approximated by polynomials. Moreover, the purely numerical part of the method is confined to the evaluation of N rN