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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Numerical calculation of integrals with strongly oscillating integrand


Authors: A. I. van de Vooren and H. J. van Linde
Journal: Math. Comp. 20 (1966), 232-245
MSC: Primary 65.55; Secondary 65.25
DOI: https://doi.org/10.1090/S0025-5718-1966-0192644-8
MathSciNet review: 0192644
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Abstract: In this paper a method is presented for evaluating

$\displaystyle \int_0^N {f(x){e^{iwx}}} dx{\text{ where }}\omega N = p \cdot 2\pi ,{\text{ }}p{\text{ integer}}{\text{.}}$

The idea is to approximate $ f(x)$ instead of the whole integrand by aid of polynomials. 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 $ \omega $. 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.


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DOI: https://doi.org/10.1090/S0025-5718-1966-0192644-8
Article copyright: © Copyright 1966 American Mathematical Society