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The Computer Journal 1976 19(3):258-267; doi:10.1093/comjnl/19.3.258
© 1976 by British Computer Society
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The use of Chebyshev series for the evaluation of oscillatory integrals

A. Alaylioglu *, G. A. Evans * and J. Hyslop *

Department of Mathematics, University of Technology, Loughborough, UK

Clenshaw and Curtis (1960) have given a scheme for the numerical integration of a well-behaved function f(x), with the interval of integration normalised to [–1, 1], which is based on the approximation of f(x) in a series of Chebyshev polynomials, Tn(x). In this context, the function is said to be well-behaved if the coefficients in the Chebyshev expansion fall off rapidly. This method is extended to integrals of the form

[equation: see PDF]

A new algorithm is presented which evaluates the resulting basic integrals directly by a method which is analogous to the automatic generation of quadrature formulae of the Newton-Cotes type, as presented by Alaylioglu, Evans and Hyslop (1975). The stability of the method is discussed and critical comparisons, including numerical tests on several practical examples, are carried out with the related earlier work of Bakhvalov and Vasil'eva (1968) and Piessens and Poleunis (1971)

Received July 1974.

* Department of Mathematics, University of Technology, Loughborough, Leicestershire LE11 3TU


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