Dynamical Behavior for Optimal Cubic-Order Multiple Solver

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1 Applied Mathematical Sciences, Vol., 7, no., 5 - HIKARI Ltd, Dynamical Behavior for Optimal Cubic-Order Multiple Solver Young Hee Geum Department of Applied Mathematics, Dankook University Cheonan, Korea -74 Copyright c 6 Young Hee Geum. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we focus on the optimal cubic-order family of iterative schemes considering the basins of attraction when the multiplicity is known. In addition, we experiment with the methods to solve 4 different test equations having roots with various multiplicities. Mathematics Subject Classification: 65H5, 65H99 Keywords: cubic-order method, basins of attraction, convergence, multiple root Introduction There are relatively few iterative methods to find the multiple roots of nonlinear equations when the multiplicity is known[],[],[],[4]. To guarantee the convergence of an iterative method, it is one of the most important things to take a good initial value close to the desired zero of the given nonlinear equation [6],[7]. Here we focus on the optimal cubic-order family of iterative schemes considering the basins of attraction[9],[7]. Let f : C C have a multiple root α with integer multiplicity m and be analytic in a small neighborhood of α. Assume that β is a fixed point of f. Then the basin of attraction of β consists of all x such that f [n] (x) β as n increases without bound, where f [n] is the nth iterate of f [],[4]. We

2 6 Young Hee Geum find an approximated α by a scheme x n+ = g(x n ), n =,,,, where g : C C is an iteration function and x C is given[8],[5]. { af(x xn+ = x n n), {f(x n) f( x n)}f (x n) x n = x n b f(xn), () f (x n) where controlled parameters a and b are to be chosen for third order of convergence[5]. where λ = m t m { xn+ = x n λ f(xn µh(xn)) f (x n), h(x n ) = f(xn) f (x n), () and µ = m( t) are controlled parameters[]. { xn+ = x n f(xn κh(xn))+γf(xn) f (x n), h(x n ) = f(xn) f (x n), () where κ = m( t) and γ = m t m are controlled parameters to be selected to guarantee the third order of convergence. Numerical examples We describe the dynamical behavior of iterative methods (), () and () and experiment with the methods to solve 4 different test equations having roots with various multiplicities[],[],[],[4]. The numerical results such as the average number of iterations per point, the number of divergent points and CPU time are displayed in Table. In all the cases, a 6 by 6 square region is centered at the origin including all the zeros of the test polynomial functions. A 6 6 uniform grid in the square is taken to mark initial points for the iterative methods to illustrate basins of attraction. Each grid point of a square is colored diversely according to the iteration number for convergence and the root it converged to α. The point is colored in black if the method do not converge after 4 iterations. We can figure out if the method converged within the maximum number of iteration allowed and if it converged to the root closer to the initial grid point[6],[7]. For plotting the complex dynamics of (), () and () with the desired basins of attraction, we take various polynomials having multiple roots with multiplicity m =, 4, 5, 6. Statistical data for the basins of attraction are tabulated in Table. In this table, abbreviations CPU, TCON, AVG and

3 Dynamical behavior for optimal cubic-order multiple solver 7 TDIV denote the value of CPU time for convergence, the value of total convergent points, the value of average iteration number for convergence and the value of divergent points. In the first instance, we have taken the following polynomial P (z) = (z + z ) whose roots z =.464 ±.654i,.688 are all with multiplicity m =. Based on Table and Figure, we realize that the method is better in view of AVG and is better in view of TDIV. As can be seen in Figure, the method has shown considerable amount of black point. These points causing divergence behavior were expected from the last column of Table. As our next sample, the polynomial P (z) = (z z) 4 has the roots z =, of multiplicity m = 4. The results are listed in Table and Figure. The method is best in veiew of CPU and TDIV. As can be seen in Figure, has shown considerable amount of black point. As the third example, we choose the polynomial P (z) = (z + 5) 5 with roots z = ±.67i and multiplicity m = 5. The results are listed in Table and Figure. The method is best in view of CPU and TDIV. The worst result for CPU is by. In the last example, we use the following polynomial P 4 (z) = (z + z + 5) 6 whose roots z =.5 ±.658i are with multiplicity m = 6. The results are presented in Table and Figure 4. The method is best in view of CPU and TDIV.

4 8 Young Hee Geum Table : Typical Examples Pm P P P P4 METHOD CPU TCON AVG TDIV Figure : The left for, the center for and the right for, for the roots of the polynomial P (z) = (z + z ) Figure : The left for, the center for and the right for, for the roots of the polynomial P (z) = (z z)4.

5 Dynamical behavior for optimal cubic-order multiple solver 9 Figure : The left for, the center for and the right for, for the roots of the polynomial P (z) = (z + 5) 5. Figure 4: The left for, the center for and the right for, for the roots of the polynomial P 4 (z) = (z + z + 5) 6.

6 Young Hee Geum Acknowledgements.The author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project Number: NRF-5RDAA88). References [] G. Adomian, Solving Frontier Problem of Physics: The Decomposition method, Kluwer Academic Publishers, Dordrecht, 994. [] R. Behl, A. Cordero, S.S. Motsa, J. R. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics, Applied Mathematics and Computation, 65 (5), [] T. Carvalho, M. Teixeira, Basin of attraction of a cusp-fold singularity in D piecewise smooth vector fileds, Journal of Mathematical Analysis and Applications, 48 (4), -. [4] C. Chun, B. Neta, Basins of attraction for several optimal fourth order methods for multiple roots, Mathematics and Computers in Simulation, (4), [5] Y.H. Geum, Y.I. Kim, Cubic convergence of parameter-controlled Newton-secant method for multiple zeros, Journal of Computational and Applied Mathematics, (9), [6] Y.H. Geum, Y.I. Kim, A two-parameter family of fourth-order iterative methods with optimal convergence for multiple zeros, Journal of Applied Mathematics, (), [7] Y.H. Geum, Y.I. Kim, A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points, Applied Mathematics and Computation, 8 (6), [8] D. Gulick, Encounters with Chaos, McGraw Hill Book, 99. [9] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, Journal of the ACM, (974),

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