Basins of Attraction for Optimal Third Order Methods for Multiple Roots

Transcription

1 Applied Mathematical Sciences, Vol., 6, no., HIKARI Ltd, Basins of Attraction for Optimal Third Order Methods for Multiple Roots 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 We compare optimal third order iterative methods illustrating the basins of attraction when the multiplicity is known. In this paper, comparisons of these methods are grounded on the number of iterations required for convergence, the divergent point and amount of CPU time. Mathematics Subject Classification: 65H5, 65H99 Keywords: third-order method, basins of attraction, convergence, nonlinear equation, multiple-root Introduction Many researchers [], [], [] have plotted the basins of attraction for the multiple zeros of nonlinear equations when the multiplicity is known. To ensure the convergence of an iterative method in a root-finding problem, it is an essential factor to take a good initial value close to the desired zero of the given nonlinear equation [6]. In this paper, we compare optimal third order iterative methods to solve nonlinear equations having roots of multiplicity m Assume that a function f : C C has a multiple root α with integer multiplicity m and is analytic in a small neighborhood of α. We 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.

2 584 Young Hee Geum LM where λ = m t m convergence. MG { xn+ = x n λ f(xn µh(xn)) f (x n), h(x n ) = f(xn) f (x n), () and µ = m( t) are parameters to be chosen for third order of { xn+ = x n f(xn µh(xn))+γf(xn) f (x n), h(x n ) = f(xn) f (x n), () where µ = m( t) and γ = m t m are controlled parameters to be selected to guarantee the third order of convergence [5]. Preliminary Notes Definition. Let e n = x n α be the error in the nth iterative step and f : R R be an iteration function with a zero α, which defines the iterative process x k+ = f(x k ). If there exist a real number p and a nonzero constant b f(x such that lim k ) α n x k = b then p is called the order of convergence and b α p is the asymptotic error constant [7], [9], []. Definition. Assume that p is a fixed point of f. Then the basin of attraction[] of p consists of all x such that f [n] (x) p as n increases without bound, where f [n] is the nth iterate of f. Numerical examples We describe the dynamical behavior[] of iterative methods () and (). Selecting an initial value close to a zero α is important to confirm the convergence of iterative function. However, it is a hard question to judge how close the initial values are to a zero α. An effective and proper way of employing stable initial values is to utilize visual basins of attraction. Considering the area of convergence on the basins of attraction, the larger area of convergence would imply a better method. Therefore, there is the need to measure size of area of convergence [4], []. Consequently, Table is shown featuring statistical data for the average number of iterations per point and the number of divergent points including CPU time. In all the cases, a 6 by 6 square region is centered at the origin and covering all the zeros of the test polynomial functions. A 6 6 uniform grid in the square is taken to unfold initial points for the iterative methods via

3 Basins of attraction for optimal third order methods 585 basins of attraction. Each grid point of a square is colored according to the iteration number for convergence and the root it converged to α. We can find 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. For plotting the complex dynamics of () and () with the desired basins of attraction, we take various polynomials having multiple roots with multiplicity m =,,, 7. Statistical data for the basins of attraction are tabulated in Table. In this table, abbreviations CPU, TCON, AVG and 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 5 7) whose roots z =.99±.86748i,.4569±.454i, are all with multiplicity m =. Based on Table and Figure, we realize that LM is better in view of AVG and TDIV. As can be seen in Figure, MG 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 5) has the roots z = ±.655i,.5598 of multiplicity m =. The results are listed in Table and Figure. The method LM performs best in view of AVG and TDIV. As the third example, we choose the polynomial P (z) = (z + z π/) whose roots z = ±.6895 are all real with multiplicity m =. The results are listed in Table and Figure. The method LM is best in view of AVG and TDIV. The best result for CPU is by LM and the worst one is by MG. As the fourth example, the experimental results of polynomial P 4 (z) = (z + z) 4 with roots z =, ± i and multiplicity m = 4 are shown in Table. From Figure 4, TDIV is for the method LM. As the fifth example, we take the polynomial P 5 (z) = (z ) 5 whose roots z = ±.75 are all real with multiplicity m = 5. The results are listed in Table and Figure 5. The method LM is better in view of CPU and AVG. As the sixth example, the results of test polynomial P 6 (z) = (z z) 6 having roots z =, with multiplicity m = 6 are listed in Table. The method LM is better in view of CPU and AVG. As can be seen in Figure 6, MG has shown a few black points. In the last example, we use the following polynomial P 7 (z) = (z z ) 7 whose roots z =.78,.78 are all real with multiplicity m = 7. The results are presented in Table and Figure 7. The method LM is best in view

4 586 Young Hee Geum Table : Typical Examples P m METHOD CPU TCON AVG TDIV P LM MG P LM MG P LM MG P 4 LM MG P 5 LM MG P 6 LM MG P 7 LM MG of AVG and TDIV. The better result for CPU is by LM and the worse one is by MG. 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: 5- RDAA-88).

5 587 Basins of attraction for optimal third order methods Figure : The left for LM, and the right for MG, for the roots of the polynomial P (z) = (z 5 7) Figure : The left for LM, and the right for MG, for the roots of the polynomial P (z) = (z + z 5) Figure : The left for LM, and the right for MG, for the roots of the polynomial P (z) = (z + z π/).

6 588 Young Hee Geum Figure 4: The left for LM, and the right for MG, for the roots of the polynomial P4 (z) = (z + z) Figure 5: The left for LM, and the right for MG, for the roots of the polynomial P5 (z) = (z ) Figure 6: The left for LM, and the right for MG, for the roots of the polynomial P6 (z) = (z z)6.

7 Basins of attraction for optimal third order methods 589 Figure 7: The left for LM, and the right for MG, for the roots of the polynomial P 7 (z) = (z z ) 7. References [] S. Amat, S. Busquier, S. Plaza, Dynamics of a family of third-order iterative methods that do not require using second derivatives, Applied Mathematics and Computation, 54 (4), [] R. Behl, A. Cordero, S.S. Motsa, J. R. Torregrosa, On developing fourthorder 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, (),

8 59 Young Hee Geum [7] D. Gulick, Encounters With Chaos, McGraw Hill, 99. [8] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, Journal of the ACM, (974), [9] J. L. Morris, Computational Methods in Elementary Numerical Analysis, John Wiley & Sons, 98. [] B. Neta, M. Scott, C. Chun, Basin attractors for various methods for multiple roots, Applied Mathematics and Computation, 8 (), [] B. Kalantari, Y. Jin, On Extraneous Fixed-points of the Basic Family of Iteration Functions, BIT Numerical Mathematics, 4 (), [] B. V. Shabat, Introduction to Complex Analysis PART II, Functions of Several Variables, American Mathematical Society, 99. [] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New Jersey, N.J., 964. Received: February, 6; Published: February, 6