This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.
References
[1]
Traub, J.F. (1977) Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York.
[2]
Sharma, J.R. (2005) A Composite Third Order Newton-Steffenson Method for Solving Nonlinear Equations. Applied Mathematics and Computation, 169, 242-246. https://doi.org/10.1016/j.amc.2004.10.040
[3]
Ostrowski, A. (1960) Solution of Equations and Systems of Equations. Academic Press, New York.
[4]
Petković, M. (2009) On Optimal Multipoint Methods for Solving Nonlinear Equations. Novi Sad Journal of Mathematics, 39, 123-130.
[5]
Neta, B. (1979) A Sixth-Order Family of Methods for Nonlinear Equations. International Journal of Computer Mathematics, 7, 157-161. https://doi.org/10.1080/00207167908803166
[6]
Khattri, S.K. and Argyros, I.K. (2010) How to Develop Fourth and Seventh Order Iterative Methods. Novi Sad Journal of Mathematics, 40, 61-67.
[7]
Bi, W.H., Ren, H.M. and Wu, Q.B. (2009) Three-Step Iterative Methods with Eighth-Order Convergence for Solving Nonlinear Equations. Journal of Computational and Applied Mathematics, 225, 105-112. https://doi.org/10.1016/j.cam.2008.07.004
[8]
Fried, I. (2013, May) High-Order Iterative Bracketing Methods. International Journal for Numerical Methods in Engineering, 94, 708-714. https://doi.org/10.1002/nme.4467
[9]
Petković, M.S., Petković, L.D. and Džunić, J. (2010) Accelerating Generators of Iterative Methods for Finding Multiple Roots of Nonlinear Equations. Computers and Mathematics with Applications, 59, 2784-2793. https://doi.org/10.1016/j.camwa.2010.01.048
[10]
Osada, N. (1994) An Optimal Multiple Root-Finding Method of Order Three. Journal of Computational and Applied Mathematics, 51, 131-133. https://doi.org/10.1016/0377-0427(94)00044-1
[11]
Chun, C., Bae, H.J. and Neta, B. (2009) New Families of Nonlinear Third-Order Solvers for Finding Multiple Roots. Computers and Mathematics with Applications, 57, 1574-1582. https://doi.org/10.1016/j.camwa.2008.10.070
[12]
Beesack, P.R. (1966) A General Form of the Remainder in Taylor’s Theorem. The American Mathematical Monthly, 73, 131-133. https://doi.org/10.2307/2313928
[13]
Poffald, E.I. (1990) The Remainder in Taylor’s Formula. The American Mathematical Monthly, 97, 205-213. https://doi.org/10.1080/00029890.1990.11995575
[14]
Fried, I. (2014) Effective High-Order Iterative Methods via the Asymptotic Form of the Taylor-Lagrange Remainder. Journal of Applied Mathematics, Article ID: 108976. https://doi.org/10.1155/2014/108976
[15]
Fried, I. (2014) The Systematic Formation of High-Order Iterative Methods. Journal of Applied and computational Mathematics, 3, 1000165. https://doi.org/10.4172/2168-9679.1000165
[16]
Liu, X. and Pan, Y. (2020) The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions. Journal of Applied Mathematics and Physics, 8, 470-480. https://doi.org/10.4236/jamp.2020.83037
[17]
Lin, G., Gao, Y. and Sun, Y. (2017) On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation. International Journal of Modern Nonlinear Theory and Application, 6, 11-25. https://doi.org/10.4236/ijmnta.2017.61002
