Chebyshev–Halley methods for analytic functions

AbstractTwo modifications of the family of Chebyshev–Halley methods are given. The first is to improve the rate of convergence to a multiple zero of an analytic function. The second is to find simultaneously all distinct zeros of a polynomial

Polynomiography for the Polynomial Infinity Norm via Kalantari's Formula and Nonstandard Iterations

In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler-Schroder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari's recent results in finding the maximum modulus of a complex polynomial based on Newton's process with the Picard iteration to other MMP-processes with various non-standard iterations

A family of root-finding methods with accelerated convergence

AbstractA parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given

Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

AbstractIn this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton’s method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frëchet derivative satisfies the Hölder continuity condition. This condition is mild and works for problems in which the second Frëchet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α∈[0,1] for the solution x∗ is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Hölder continuity condition on F″, it is found that our analysis gives improved results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Chebyshev method (α=0) and the convex acceleration of Newton’s method (α=1) respectively with improved results

Semilocal convergence of a family of iterative methods in Banach spaces

[EN] In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory.This work has been supported by Ministerio de Ciencia e Innovaci´on MTM2011-28636-C02-02 and by Vicerrectorado de Investigaci´on. Universitat Polit`ecnica de Val`encia PAID-SP-2012-0498Hueso Pagoaga, JL.; Martínez Molada, E. (2014). Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms. 67(2):365-384. https://doi.org/10.1007/s11075-013-9795-7S365384672Traub, J.F.: Iterative Methods for the Solution of Nonlinear Equations. Prentice Hall, New York (1964)Kantorovich, L.V.: On the newton method for functional equations. Doklady Akademii Nauk SSSR 59, 1237–1240 (1948)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, I: The Halley method. Computing 44, 169–184 (1990)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, II: The Chebyshev method. Computing 45, 355–367 (1990)Hernández, M.A.: Reduced recurrence relations for the Chebyshev method. J. Optim. Theory Appl. 98, 385–397 (1998)Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for super-Halley method. J. Comput. Math. Appl. 7, 1–8 (1998)Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-like methods. Appl. Math. Optim. 41, 227–236 (2000)Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)Argyros, I., K., Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Hilout, S.: On the semilocal convergence of efficient Chebyshev Secant-type methods. J. Comput. Appl. Math. 235–10, 3195–3206 (2011)Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newtons method. J. Complex. 28(3), 364–387 (2012)Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algoritm. 54, 497–516 (2011)Kou, J., Li, Y., Wang, X.: A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)Zheng, L., Gu, C.: Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algoritm. 59, 623–638 (2012)Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algoritm. 57, 441–456 (2011)Hernández, M.A.: The newton method for operators with hlder continuous first derivative. J. Optim. Appl. 109, 631–648 (2001)Ye, X., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Hlder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006)Zhao, Y., Wu, Q.: Newton-Kantorovich theorem for a family of modified Halleys method under Hlder continuity conditions in Banach spaces. Appl. Math. Comput. 202, 243–251 (2008)Argyros, I.K.: Improved generalized differentiability conditions for Newton-like methods. J. Complex. 26, 316–333 (2010)Hueso, J.L., Martínez. E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)Taylor, A.Y., Lay, D.: Introduction to Functional Analysis, 2nd edn.New York, Wiley (1980)Jarrat, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)Cordero, A., Torregrosa, J.R.: Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

High order iterative methods for decomposition‐coordination problems

Many real‐life optimization problems are of the multiobjective type and highdimensional. Possibilities for solving large scale optimization problems on a computer network or multiprocessor computer using a multi‐level approach are studied. The paper treats numerical methods in which procedural and rounding errors are unavoidable, for example, those arising in mathematical modelling and simulation. For the solution of involving decomposition‐coordination problems some rapidly convergent interative methods are developed based on the classical cubically convergent method of tangent hyperbolas (Chebyshev‐Halley method) and the method of tangent parabolas (Euler‐Chebyshev method). A family of iterative methods having the convergence order equal to four is also considered. Convergence properties and computational aspects of the methods under consideration are examined. The problems of their global implementation and polyalgorithmic strategy are discussed as well. Daugialaipsniai iteraciniai metodai skaidymo ir jungimo problemoms spręsti Santrauka Daugelis realių optimizavimo uždavinių yra daugiatiksliai ir daugiadimensiai. Straipsnyje nagrinėjamos sudėtingų optimizacijos uûdavinių sprendimų galimybės daugialaipsniu metodu, naudojant kompiuterinį tinklą arba daugiaprocesorinį kompiuterį. Apžvelgiami tokie įprastininiai skaitiniai metodai, kaip matematinis modeliavimas, kuriame neiövegiama paklaidų apvalinimo. Skaidymo ir jungimo problemoms spręsti, remiantis tangentinės hiperbolės (Čebyševo ir Halėjaus) ir tangentinės parabolės (Oilerio ir Čebyševo) metodais, sukurti keli greitos konvergencijos interaciniai metodai. Taip pat aptariama ketvirtojo laipsnio konvergencijos metodų šeima. Nagrinėjamos sukurtųjų metodų konvergavimo savybės ir skaičiavimo jais aspektai. Svarstomos pasiūlytųjų metodų ir polialgoritminės strategijos visuotinio taikymo galimybės. First Published Online: 21 Oct 2010 Reikšminiai žodžiai: Banacho erdvė, daugiatikslė optimizacija, hierachinis sprendimų priėmimas, skaidymo ir jungimo schemos, tangentinės hiperbolės ir tangentinės parabolės metodai, globalinė konvergencija

Parallel schemes for global interative zero-finding.

by Luk Wai Shing.Thesis (M.Phil.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves 44-45).ABSTRACT --- p.iACKNOWLEDGMENTS --- p.iiChapter CHAPTER 1. --- INTRODUCTION --- p.1Chapter CHAPTER 2. --- DRAWBACKS OF CLASSICAL THEORY --- p.4Chapter 2.1 --- Review of Sequential Iterative Methods --- p.4Chapter 2.2 --- Visualization Techniques --- p.8Chapter 2.3 --- Review of Deflation --- p.10Chapter CHAPTER 3. --- THE IMPROVEMENT OF THE ABERTH METHOD --- p.11Chapter 3.1 --- The Durand-Kerner method and the Aberth method --- p.11Chapter 3.2 --- The generalized Aberth method --- p.13Chapter 3.3 --- The modified Aberth Method for multiple-zero --- p.13Chapter 3.4 --- Choosing the initial approximations --- p.15Chapter 3.5 --- Multiplicity estimation --- p.16Chapter CHAPTER 4. --- THE HIGHER-ORDER ITERATIVE METHODS --- p.18Chapter 4.1 --- Introduction --- p.18Chapter 4.2 --- Convergence analysis --- p.20Chapter 4.3 --- Numerical Results --- p.28Chapter CHAPTER 5. --- PARALLEL DEFLATION --- p.32Chapter 5.1 --- The Algorithm --- p.32Chapter 5.2 --- The Problem of Zero Component --- p.34Chapter 5.3 --- The Problem of Round-off Error --- p.35Chapter CHAPTER 6. --- HOMOTOPY ALGORITHM --- p.36Chapter 6.1 --- Introduction --- p.36Chapter 6.2 --- Choosing Q(z) --- p.37Chapter 6.3 --- The arclength continuation method --- p.38Chapter 6.4 --- The bifurcation problem --- p.40Chapter 6.5 --- The suggested improvement --- p.41Chapter CHAPTER 7. --- CONCLUSION --- p.42REFERENCES --- p.44APPENDIX A. PROGRAM LISTING --- p.A-lAPPENDIX B. COLOR PLATES --- p.B-

Higher Order Methods of the Basic Family of Iterations via S-Iteration Scheme with s-Convexity

There are many methods for solving a polynomial equation and many different modifications of those methods have been proposed in the literature. One of such modifications is the use of various iteration processes taken from the fixed point theory. In this paper, we propose a modification of the iteration processes used in the Basic Family of iterations by replacing the convex combination with an s-convex one. In our study, we concentrate only on the S-iteration with s-convexity. We present some graphical examples, the so-called polynomiographs, and numerical experiments showing the dependency of polynomiograph’s generation time on the value of the s parameter in the s-convex combination