Abstract
Higher-order methods for the simultaneous inclusion of complex zeros of algebraic polynomials are presented in parallel (total-step) and serial (single-step) versions. If the multiplicities of each zeros are given in advance, the proposed methods can be extended for multiple zeros using appropriate corrections. These methods are constructed on the basis of the zero-relation of Gargantini’s type, the inclusion isotonicity property and suitable corrections that appear in two-point methods of the fourth order for solving nonlinear equations. It is proved that the order of convergence of the proposed methods is at least six. The computational efficiency of the new methods is very high since the acceleration of convergence order from 3 (basic methods) to 6 (new methods) is attained using only n polynomial evaluations per iteration. Computational efficiency of the considered methods is studied in detail and two numerical examples are given to demonstrate the convergence behavior of the proposed methods.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic, New York (1983)
Brent, R.P.: Some efficient algorithms for solving systems of nonlinear equations. SIAM J. Numer. Anal. 10, 327–344 (1973)
Brent, R.P.: Multi-precision zero-finding methods and the complexity of elementary function evaluation. In: Traub, J.F. (ed.) Analytic Computational Complexity, pp. 151–176. Academic, New York (1975) (Reprinted with minor corrections in 1999)
Brent, R., Zimmermann, P.: Modern Computer Arithmetic. Cambridge University Press, Cambridge (2011)
Carstensen, C.: Anwendungen von Begleitmatrizen. Z. Angew. Math. Mech. 71, 809–812 (1991)
Carstensen, C., Petković, M.S.: An improvement of Gargantini’s simultaneous inclusion method for polynomial roots by Schroeder’s correction. Appl. Numer. Math. 13, 453–468 (1994)
Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33, 1–15 (2007)
Fujimoto, J., Ishikawa, T., Perret-Gallix, D.: High Precision Numerical Computations. Technical report, ACCP-N-1 (2005)
Gargantini, I.: Further application of circular arithmetic: Schröder-like algorithms with error bound for finding zeros of polynomials. SIAM J. Numer. Anal. 15, 497–510 (1978)
Gargantini, I., Henrici, P.: Circular arithmetic and the determination of polynomial zeros. Numer. Math. 18, 305–320 (1972)
Granlund, T.: GNU MP; The GNU Multiple Precision Arithmetic Library, Edition 5.0.0 (2010)
Herceg, \(\mbox{\raise0.3ex\hbox{-}\kern-0.4em D}\).D.: Computer Implemention and Interpretation of Iterative Methods for Solving Equations. Master thesis, University of Novi Sad, Novi Sad (1997)
Herzberger, J., Metzner, L.: On the Q-order and R-order of convergence for coupled sequences arising in iterative numerical processes. In: Alefeld, G., Herzberger, J. (eds.) Numerical Methods and Error Bounds Mathematical Research, vol. 89, pp. 120–131. Akademie, Berlin (1996)
Kravanja, P.: On Computing Zeros of Analytic Functions and Related Problems in Structured Numerical Linear Algebra. Ph.D. thesis, Katholieke Universiteit Leuven, Lueven (1999)
Kravanja, P.: A modification of Newton’s method for analytic mappings having multiple zeros. Comput. 62, 129–145 (1999)
Li, S., Liao, X., Cheng, L.: A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)
McNamee, J.M.: Numerical Methods for Roots of Polynomials, Part I. Elsevier, Amsterdam (2007)
Neumaier, A.: An existence test for root clusters and multiple roots. Z. Angew. Math. Mech. 68, 256–257 (1988)
Niu, X.M., Sakurai, T.: A method for finding the zeros of polynomials using a companion matrix. Jpn. J. Ind. Appl. Math. 20, 239-256 (2003)
Ortega, J.M., Rheiboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)
Petković, M.S.: On a generalization of the root iterations for polynomial complex zeros in circular interval arithmetic. Computing 27, 37–55 (1981)
Petković, M.S.: Iterative Methods for Simultaneous Inclusion of Polynomial Zeros. Springer, Berlin (1989)
Petković, M.S.: On the Halley-like algorithms for the simultaneous approximation of polynomial complex zeros. SIAM J. Numer. Anal. 26, 740–763 (1989)
Petković, M.S.: The self-validated method for polynomial zeros of high efficiency. J. Comput. Appl. Math. 233, 1175–1186 (2009)
Petković, M.S., Carstensen, C.: On some improved inclusion methods for polynomial roots with Weierstrass’ correction. Comput. Math. Appl. 25, 59–67 (1993)
Petković, M.S., Carstensen, C., Trajković, M.: Weierstrass’ formula and zero-finding methods. Numer. Math. 69, 353–372 (1995)
Petković, M.S., Milošević, D.: Improved Halley-like methods for the inclusion of polynomial zeros. Appl. Math. Comput. 169, 417–436 (2005)
Petković, M.S., Milošević, D.M.: On a new family of simultaneous methods with corrections for the inclusion of polynomial zeros. Int. J. Comput. Math. 83, 299–317 (2006)
Petković, M.S., Petković, L.D.: Complex Interval Arithmetic and its Applications. Wiley, Berlin (1998)
Schönhage, A., Strassen, V.: Schnelle multiplication grosser Zahlen. Computing 7, 281–292 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Serbian Ministry of Science under grant number 174022.
Rights and permissions
About this article
Cite this article
Petković, M.S., Milošević, M.R. & Milošević, D.M. New higher-order methods for the simultaneous inclusion of polynomial zeros. Numer Algor 58, 179–201 (2011). https://doi.org/10.1007/s11075-011-9452-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-011-9452-y