A new quadrature method for locating the zeros of analytic functions with applications to engineering problems
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Ioakimidis, Nikolaos
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Abstract
A new method for the computation of real or complex zeros of analytic functions and/or poles of meromorphic functions outside a fundamental interval [a,b] of the real axis is proposed. This method is based on appropriately taking into account the error terms in the Gauss– and Lobatto–Chebyshev quadrature rules for ordinary integrals and it leads to a very simple non-iterative algorithm for the computation of these zeros. The results obtained by this algorithm with very few functional evaluations are of a very good accuracy and they can further be improved, if required, by local methods, which are generally inappropriate for the original localization of the zeros. The proposed method was tested in two engineering problems: a problem of neutron moderation in nuclear reactors and a problem of determining the critical buckling load of an elastic frame. The corresponding transcendental equations were solved by this method and numerical results for their zeros are presented. In all equations solved, numerical values for their zeros accurate to at least five significant digits were obtained by the present method with no more than thirteen functional evaluations.
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Analytic functions, Meromorphic functions, Transcendental functions, Zeros, Poles, Chebyshev polynomials, Chebyshev functions of the second kind, Numerical integration, Gauss– and Lobatto–Chebyshev quadrature rules, Convergence, Error term, Quadrature error, Neutron moderation, Elastic buckling