On the local convergence of Gargantini-Farmer-Loizou method for simultaneous approximation of multiple polynomial zeros

Volume 11, Issue 9, pp 1045--1055 http://dx.doi.org/10.22436/jnsa.011.09.03

Publication Date: June 19, 2018 Submission Date: April 15, 2018 Revision Date: May 04, 2018 Accteptance Date: May 12, 2018

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Authors

Petko D. Proinov - Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria.

Abstract

The paper deals with a well known iterative method for simultaneous computation of all zeros (of known multiplicities) of a polynomial with coefficients in a valued field. This method was independently introduced by Farmer and Loizou [M. R. Farmer, G. Loizou, Math. Proc. Cambridge Philos. Soc., \({\bf 82}\) (1977), 427--437] and Gargantini [I. Gargantini, SIAM J. Numer. Anal., \({\bf 15}\) (1978), 497--510]. If all zeros of the polynomial are simple, the method coincides with the famous Ehrlich's method [L. W. Ehrlich, Commun. ACM, \({\bf 10}\) (1967), 107--108]. We provide two types of local convergence results for the Gargantini-Farmer-Loizou method. The first main result improves the results of [N. V. Kyurkchiev, A. Andreev, V. Popov, Ann. Univ. Sofia Fac. Math. Mech., \({\bf 78}\) (1984), 178--185] and [A. I. Iliev, C. R. Acad. Bulg. Sci., \({\bf 49}\) (1996), 23--26] for this method. Both main results of the paper generalize the results of Proinov [P. D. Proinov, Calcolo, \({\bf 53}\) (2016), 413--426] for Ehrlich's method. The results in the present paper are obtained by applying a new approach for convergence analysis of Picard type iterative methods in finite-dimensional vector spaces.

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Keywords

• Iterative methods
• simultaneous methods
• Ehrlich method
• multiple polynomial zeros
• Gargantini-Farmer-Loizou method
• local convergence
• error estimates

MSC

•  65H05
•  47H09
•  47H17
•  12Y05

References

• [1] L. W. Ehrlich, A modified Newton method for polynomials, Commun. ACM, 10 (1967), 107–108.
  • View Article
  • Google Scholar


• [2] A. J. Engler, A. Prestel, Valued Fields, Springer Monographs in Mathematics, Berlin (2005)
  • View Article
  • Google Scholar


• [3] M. R. Farmer, G. Loizou, An algorithm for the total or partial factorization of a polynomial, Math. Proc. Cambridge Philos. Soc., 82 (1977), 427–437.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] I. Gargantini, Further applications of circular arithmetic: Schröder-like algorithms with error bounds for finding zeros of polynomials, SIAM J. Numer. Anal., 15 (1978), 497–510
  • View Article
  • MathSciNet
  • Google Scholar


• [5] A. I. Iliev, A generalization of Obreshkoff-Ehrlich method for multiple roots of polynomial equations, C. R. Acad. Bulg. Sci., 49 (1996), 23–34.
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. I. Iliev, A generalization of Obreshkoff-Ehrlich method for multiple roots of algebraic, trigonometric and exponential equations, Math. Balkanica, 14 (2000), 17–28.
  • MathSciNet
  • Google Scholar
  • MATH


• [7] S. I. Ivanov, A unified semilocal convergence analysis of a family of iterative algorithms for computing all zeros of a polynomial simultaneously, Numer. Algorithms, 75 (2017), 1193–1204.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] N. V. Kyurkchiev, A. Andreev, Ehrlich’s methods with a raised speed of convergence, Serdica Math. J., 13 (1987), 52–57.
  • Google Scholar
  • MATH


• [9] N. V. Kyurkchiev, A. Andreev, V. Popov, Iterative methods for computation of all multiple roots an algebraic polynomial, Ann. Univ. Sofia Fac. Math. Mech., 78 (1984), 178–185.
  • MATH


• [10] G. V. Milovanović, M. S. Petković, On the convergence order of a modified method for simultaneous finding polynomial zeros, Computing, 30 (1983), 171–178.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] G. V. Milovanović, M. S. Petković, On computational efficiency of the iterative methods for simultaneous approximation of polynomial zeros, ACM Trans. Math. Soft., 12 (1986), 295–306
  • View Article
  • Google Scholar


• [12] M. S. Petković, G. V. Milovanović, A note on some improvements of the simultaneous method for determination of polynomial zeros, J. Comput. Appl. Math., 9 (1983), 65–69.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] M. S. Petković, G. V. Milovanović, L. V. Stefanović, On the convergence order of an accelerated simultaneous method for polynomial complex zeros, Z. Angew. Math. Mech., 66 (1986), 428–429.
  • MathSciNet
  • Google Scholar


• [14] M. S. Petković, G. V. Milovanović, L. V. Stefanović, Some higher-order methods for the simultaneous approximation of multiple polynomial zeros, Comput. Math. Appl. Ser. A, 12 (1986), 951–962.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] M. S. Petković, B. Neta, L. D. Petković, J. Džunić, Multipoint methods for solving nonlinear equations, Elsevier/ Academic Press, Amsterdam (2013)
  • Google Scholar


• [16] P. D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26 (2010), 3–42.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] P. D. Proinov, General convergence theorems for iterative processes and applications to the Weierstrass root-finding method, J. Complexity, 33 (2016), 118–144.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] P. D. Proinov, A general semilocal convergence theorem for simultaneous methods for polynomial zeros and its applications to Ehrlich’s and Dochev-Byrnev’s methods, Appl. Math. Comput., 284 (2016), 102–114.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] P. D. Proinov, On the local convergence of Ehrlich method for numerical computation of polynomial zeros, Calcolo, 53 (2016), 413–426.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] P. D. Proinov, Unified convergence analysis for Picard iteration in n-dimensional vector spaces, Calcolo, 2018 (2018), 21 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] P. D. Proinov, S. I. Cholakov, Convergence of Chebyshev-like method for simultaneous approximation of multiple polynomial zeros, C. R. Acad. Bulg. Sci., 67 (2014), 907–918.
  • MathSciNet
  • Google Scholar
  • MATH


• [22] P. D. Proinov, M. T. Vasileva, On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously, J. Inequal. Appl., 2015 (2015), 25 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] S. Tashev, N. Kyurkchiev, Certain modifications of Newton’s method for the approximate solution of algebraic equations, Serdica Bulg. Math. Publ., 9 (1983), 67–73.
  • Google Scholar


• [24] P. Tilli, Convergence conditions of some methods for the simultaneous computation of polynomial zeros, Calcolo, 35 (1998), 3–15.
  • View Article
  • MathSciNet
  • Google Scholar


• [25] D. R. Wang, F. G. Zhao, Complexity analysis of a process for simultaneously obtaining all zeros of polynomials, Computing, 43 (1989), 187–197.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH