Finding the Zeros of a High-Degree Polynomial Sequence

Vladimir L. Borsch, Peter I. Kogut


A 1-parameter initial-boundary value problem for a linear spatially 1-dimensional
homogeneous degenerate wave equation, posed in a space-time rectangle, in case of strong degeneracy, was reduced to a linear integro-differential equation of convolution type (JODEA, 29(1) (2021), pp. 1–31). The former was then solved by applying the Laplace transformation, and the solution formula was inverted in the form of the Neumann series. The current study deals with an other approach to the inversion of the solution formula, based on invoking the Bromwich integral and the Cauchy residue theorem for the integrand. The denominator of the integrand being an infinite series with respect to rational functions of the complex variable, converges quite rapidly and can be approximated with finite series of m terms. Therefore finding the zeros of the approximated denominator reduces to finding the zeroes of a polynomial of degree 2m. For the resulting polynomial sequence some numerical approaches have been applied.


Ключові слова

degenerate wave equation; linear integro-differential equation of convolution type; Laplace transformation

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V. L. Borsch, P. I. Kogut, Solutions to a simplified initial boundary value problem for 1D hyperbolic equation with interior degeneracy, Journal of Optimization, Differential Equations, and their Applications (JODEA), 29(1) (2021), 1 – 31.

J. Bremer, An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments, Advances in Computational Mathematics, 45(1) (2019), 173 – 211.

G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, NY, 1974.

D. Gottlieb, C.-W. Shu, On the Gibbs phenomenon and its resolution, SIAM Review, 39(4) (1997), 644 – 668.

J. Hubbard, S. Schleicher, S. Sutherland, How to find all roots of complex polynomials by Newton’s method, Inventiones Mathematicae, 146(1) (2001), 1 – 33.

A. J. Jerri, The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations, Springer Science+Business Media, LLC, Dordrecht, 1998.

A. J. Jerri, A Lanczos-like -factors for reducing the Gibbs phenomenon in general orthogonal expansions and other representations, Journal of Computational Analysis and Applications, 2(24) (2000), 111 – 127.

P. Kowalczyk, Complex Root Finding Algorithm Based on Delaunay Triangulation, ACM Transactions on Mathematical Software, 41(3) (2015), Article 19.

P. Kravanja, M. Van Barel, Computing the Zeros of Analytic Functions, Springer, Berlin, 2000.

C. Lanczos, Applied Analysis, Prentice Hall, Englewood Cliffs, 1956.

Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, Lectures on the Theory of Functions of a Complex Variable, Mir Publishers, Moscow, 1985.

N. M. Temme, An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives, Journal of Computational Physics, 32(2) (1979), 270 – 279.

G. P. Tolstov, Fourier Series, Dover, NY, 1962.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.



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