Abstract
A method is presented for the calculation of roots of non-polynomial functions, motivated by the requirement to generate quadrature rules based on non-polynomial orthogonal functions. The approach uses a combination of local Taylor expansions and Sturm’s theorem for roots of a polynomial which together give a means of efficiently generating estimates of zeros which can be polished using Newton’s method. The technique is tested on a number of realistic problems including some chosen to be highly oscillatory and to have large variations in amplitude, both of which features pose particular challenges to root–finding methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bornemann, F.: Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals. Found. Comput. Math. 11(1), 1–63 (2011). doi:10.1007/s10208-010-9075-z
Boyd, J.P.: Finding the zeros of a univariate equation: Proxy rootfinders, Chebyshev interpolation and the companion matrix. SIAM Rev. 55(2), 375–396 (2013). doi:10.1137/110838297
Bracciali, C.F., McCabe, J.H., Pérez, T.E., Sri Ranga, A.: A class of orthogonal functions given by a three term recurrence formula. Math. Comput. 85 (300), 1837–1859 (2016). doi:10.1090/mcom3041
Bracciali, C.F., Silva, J.S., Sri Ranga, A.: Explicit formulas for OPUC and POPUC associated with measures which are simple modifications of the Lebesgue measure. Appl. Math. Comput. 271, 820–831 (2015). doi:10.1016/j.amc.2015.09.067
Bremer, J.: On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations. SIAM J. Sci. Comput. 39(1), A55–A82 (2017). doi:10.1137/16M1057139
Bremer, J., Rokhlin, V.: Improved estimates for nonoscillatory phase functions. Discret. Contin. Dyn. Syst. 36(8), 4101–4131 (2016). doi:10.3934/dcds.2016.36.4101
Costa, M.S., Felix, H.M., Sri Ranga, A.: Orthogonal polynomials on the unit circle and chain sequences. J. Approx. Theory 173 (2013). doi:10.1016/j.jat.2013.04.009
Glaser, A., Liu, X., Rokhlin, V.: A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput. 29(4), 1420–1438 (2007). doi:10.1137/06067016X
Gradshteyn, I., Ryzhik, I.M.: Table of integrals, series, and products, 5th edn. Academic, London (1980)
Hook, D.G., McAree, P.R.: Using Sturm sequences to bracket real roots of polynomial equations. In: Glassner, A.S. (ed.) Graphics Gems. Academeic Press, London (1990)
Jones, W.B., Njåstad, O., Thron, W.J.: Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. Lond. Math. Soc. 21(2), 113–152 (1989). doi:10.1112/blms/21.2.113
Simon, B.: Orthogonal polynomials on the unit circle: Part 1. Classical theory, vol. 54. American Mathematical Society Colloqium Publications (2005)
Sturm, C.: Analyse d’un mémoire sur la résolution des équations numériques. In: Pont, J.C. (ed.) Collected Works of Charles François Sturm, pp. 345–390, Birkhäuser Basel, Basel. doi:10.1007/978-3-7643-7990-2_24 10.1007/978-3-7643-7990-2_24. Originally published 1835 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bracciali, C.F., Carley, M. Quasi-analytical root-finding for non-polynomial functions. Numer Algor 76, 639–653 (2017). https://doi.org/10.1007/s11075-017-0274-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0274-4