Abstract
We presented new two-point methods for the simultaneous approximation of all n simple (real or complex) zeros of a polynomial of degree n. We proved that the R-order of convergence of the total-step version is three. Moreover, computationally verifiable initial conditions that guarantee the convergence of one of the proposed methods are stated. These conditions are stated in the spirit of Smale’s point estimation theory; they depend only on available data, the polynomial coefficients, polynomial degree n and initial approximations \(x_{1}^{(0)},\ldots ,x_{n}^{(0)}\), which is of practical importance. Using the Gauss-Seidel approach we state the corresponding single-step version and consequently its prove that the lower bound of its R-order of convergence is at least 2 + y n > 3, where y n ∈ (1, 2) is the unique positive root of the equation y n − y − 2 = 0. Two numerical examples are given to demonstrate the convergence behavior of the considered methods, including global convergence.
Similar content being viewed by others
References
Aberth, O.: Iterative methods for finding all zeros of a polynomial simultaneously. Math. Comput. 27, 339–344 (1973)
Alefeld, G., Herzberger, J.: On the convergence speed of some algorithms for the simultaneous approximation of polynomial zeros. SIAM J. Numer. Math 11, 237–243 (1974)
Alefeld, G., Herzberger, J.: Introduction to Interval Computation. Academic, New York (1983)
Batra, P.: Improvement of a convergence condition for the Durand-Kerner iteration. J. Comput. Appl. Math. 96, 117–125 (1998)
Chen, P.: Approximate zeros of quadratically convergent algorithms. Math. Comput. 63, 247–270 (1994)
Cira, O.: Metode Numerice Pentru Rezolvarea Ecuaţiilor (in Romanian). Editura Academiei Romane, Bucureşti (2005)
Cira, O.: The Convergence of Simultaneous Inclusion Methods. Matrix Rom, Bukureşti (2012)
Loizou, G.: Higher-order iteration functions for simultaneously approximating polynomial zeros. Int. J. Comput. Math. 14, 45–58 (1983)
McNamee, J.M.: Numerical Methods for Roots of Polynomials. Part I. Elsevier, Amsterdam (2007)
Nedzhibov, G.H.: A derivative-free iterative method for simultaneously computing an arbitrary number of zeros of nonlinear equations. Comput. Math. Appl. 63, 1185–1191 (2012)
Nourein, A.W.M.: An improvement on Nourein’s method for the simultaneous determination of the zeros of a polynomial (an algorithm). J. Comput. Appl. Math. 3, 109–110 (1977)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)
Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic, New York (1960)
Petković, M.S., Carstensen, C., Trajković, M.: Weierstrass’ formula and zero-finding methods. Numer. Math. 69, 353–372 (1995)
Petković, M.S.: On initial conditions for the convergence of simultaneous root finding methods. Computing 57, 163–177 (1996)
Petković, M.S.: Point Estimation of Root Finding Methods. Springer, Berlin-Heidelberg (2008)
Petković, M.S., Džunić, J., Petković, L.D.: A family of two-point methods with memory for solving nonlinear equations. Appl. Anal. Discrete Math. 5, 298–317 (2011)
Petković, M.S., Neta, B., Petković, L.D., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier/Academic Press, Amsterdam (2013)
Petković, M.S., Petković, L.D.: Complex Interval Arithmetic and its Applications. Wiley-VCH, New York (1998)
Petković, M.S., Petković, L.D.: On acubically convergent derivative free root finding method. Int. J. Comput. Math. 84, 505–513 (2007)
Petković, M.S., Petković, L.D.: Families of optimal multipoint methods for solving nonlinear equations: a survey. Appl. Anal. Discret. Math. 4, 1–22 (2010)
Petković, M.S., Rančić, L., Petković, L.D.: Point estimation of simultaneous methods for solving polynomial equations: a survey (II). J. Comput. Appl. Math. 205, 32–52 (2007)
Petković, M.S., Stefanović, L.V.: On the convergence order of accelerated root iterations. Numer. Math. 44, 463–476 (1984)
Potra, F.A.: On Q-Order and R-Order of convergence. J. Optim. Theor. Appl. 63, 415–431 (1989)
Potra, F.A., Pták, V.: Nondiscrete Induction and Iterative Processes. Pitman, Boston (1984)
Proinov, P.D.: Semilocal convergence of two iterative methods for simultaneous computation of polynomial zeros. C. R. Acad. Bulg. Sci. 59, 705–712 (2006)
Proinov, P.D.: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex. 26, 3–42 (2010)
Smale, S.: Newton’s method estimates from data at one point. In: The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986)
Tilli, P.: Convergence conditions of some methods for the simultaneous computation of polynomial zeros. Calcolo 35, 3–15 (1998)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs New Jersey (1964)
Varga, R.S.: Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs New Jersey (1962)
Wall, D.: The order of an iteration formula. Math. Comput. 10, 167–168 (1956)
Wang, D., Zhao, F.: The theory of Smale’s point estimation and its application. J. Comput. Appl. Math. 60, 253–269 (1995)
Weierstrass, K.: Neuer Beweis des Satzes, dass jede ganze rationale Funktion einer Veränderlichen dargestellt werden kann als ein Product aus linearen Funktionen derstelben Verändelichen. Ges. Werke 3 (1903), 251–269 (1967). Johnson Reprint Corp., New York
Werner, W.: On the simultaneous determination of polynomial roots. In: Iterative Solution of Nonlinear Systems of Equations, Lecture Notes in Mathematics, Vol. 953, pp 188–202. Springer-Verlag, Berlin (1982)
Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice Hall, New Jersey (1963)
Zhao, F., Wang, D.: The theory of Smale’s point estimation and the convergence of Durand-Kerner program. Math. Numer. Sin. 2, 198–206 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Serbian Ministry of Education and Science under grant number 174022.
Rights and permissions
About this article
Cite this article
Petković, M.S., Rančić, L.Z. On the guaranteed convergence of new two-point root-finding methods for polynomial zeros. Numer Algor 67, 187–222 (2014). https://doi.org/10.1007/s11075-013-9782-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9782-z