Development of a Family of Optimal Quartic-Order Methods for Multiple Roots and Their Dynamics

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1 Applied Mathematical Sciences, Vol. 9, 5, no. 49, HIKARI Ltd, Development of a Family of Optimal Quartic-Order Methods for Multiple Roots and Their Dynamics Young Hee Geum Department of Applied Mathematics, Dankook University Cheonan, Korea -74 Copyright c 5 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 develop a family of optimal fourth-order multipleroot solvers and illustrate the complex dynamical behavior related to the basins of attraction. Numerical examples are presented to demonstrate convergence behavior agreeing with theory. Mathematics Subject Classification: 65H5, 65H99 Keywords: quartic-order method, basin of attraction, convergence, nonlinear equation, multiple-root Introduction Finding multiple roots of a scalar function f(x) = ranks among the most important problems in applied mathematics. Since Traub [] studied his work on optimal iteration theory, many researchers [],[4],[5],[8] have constructed optimal iterative schemes for finding the multiple roots of nonlinear equations. In this paper, the following iteration method free of second derivatives is proposed to find an approximate root α, given an initial guess x sufficiently Corresponding author

2 748 Young Hee Geum Table : Typical methods with selected parameters (λ, ρ) and constants (b, c, d) SN (λ, ρ) (b, c, d) Y ( m(m+4) (m+) δ, m(m+) ) b =, c =, d = m(+m) κ δ ( 4+m(+m)) Y (/, -) b = m (m(+m)( +κ) 4κ), c = m (+m) m (4+m(5+m(8+5m)))κ+8m(+m)(+m) κ 4(+m) (+m) κ 6(+m)κ 6(+m)κ d = m (+m) +m (4+m(5+m(8+5m)))κ 6m(+m)(+m) κ +(+m)(+m) 4 κ 6(+m)κ Y (m, κ ) b = m 4+m+4m κ, c = 4+m(+m) + 4m Y4 (, ) b = m (+m) 6κ Y5 m m (, ( 4+m+m )κ ) ( m(+m) 4 +m+m ) κ d = 4+m(4+m+m(4+m(+m))κ) ( m +m+m ), κ (m(+m), c = m 4m(8+m(5+m))κ+4(+m) κ ) ( κ d = m (+m) ) 4(8+m(5+m))κ, κ m(4+m) (+m) ) ( 4+m(+m))κ ( 4+m(+m)), d = m (4+m) ( 4+m(+m)) (m(+m) 4m(8+m(5+m))κ +4(+m) κ 4) κ, d = m ( (+m) 4(8+m(5+m))κ ) κ ( b =, c = m Y6 (κ, ) b = m (+m) 6κ, c = m Here, δ = κ ( m + m 4 ) close to α: y n = x n m f(x n) m+ f (x n) x n+ = y n b F (yn) cf(xn)+df (yn) f (x n) f (y n), F (y n ) = f(x n ) (y n x n ) λf (x n)f (y n) f (x n)+ρf (y n), () where b, c, d, λ and ρ are parameters to be chosen for maximal order of convergence[6],[7],[9]. Preliminary Notes Definition. Let f : R R be an iteration function with a root α, which defines the iterative process x k+ = f(x k ). Let e n = x n α be the error in the nth iterative step. If there exist a real number p and a nonzero constant b such that f(x k ) α lim = b, () n x k α p then p is called the order of convergence and b is the asymptotic error constant and the relation e n+ = be p n +O(e n ) p+ is called the error equation[6],[7],[9],[]. Definition. Assume that theoretical asymptotic error constant η = lim n e n log en/η log e n as the computa- and convergence order p are known. Define p n = tional convergence order. Note that lim n p n = p Definition. Let d be the number of distinct functional or derivative evaluations required by an iterative method per step. The efficiency index of the method is defined by EI = p /d, where p is the order of convergence. e n p

3 Iterative methods for multiple roots and their dynamics 749 Main Results Theorem. Let f : C C have a zero α with integer multiplicity m and be analytic in a small neighborhood of α. Let λ, ρ R be two free constant parameters. Let κ = ( m m+ )m, γ = m and θ (+m) j = f (m+j) (α) for j N. Let f (m) (α) x be an initial guess chosen in a sufficiently small neighborhood of α. Let b = (m (4κρ m( + m)( + κρ)))/(6mκλ 8( + m)κρ), c = (υ + κm(m + ) (4κ λρ(m(λ ρ) ρ) + κ(λ ρ)(λm (m + )ρ) + mρ))ς and d = (υ m ( + m) κ( + κλ)ρ m( + m) 4 κ ( + κλ)ρ )ς with ς = (6κλ(mλ ( + m)ρ)) and υ = m (( + m) 4(8 + m(5 + m))κλ) + ( + m) 5 κ ρ. Then iterative methods () are optimal and of order four and possess the following error equation: e n+ = ψ 4 e 4 n + O(e 5 n), where ψ 4 = m5 +6m 4 +4m +4m 4 8+m( +) κρ+κmρ+m θ m 4 (m+) (m+) θ m(m+) (m+) θ + Proof. The optimality is pursued based on Kung-Traub s conjecture[6] due to three evaluations of functions or derivatives. Hence, it suffices to determine the constant parameters for quartic-order convergence. Expanding f(x n ) about α by a Taylor s series expansion, we have m θ (m+) (m+)(m+). f(x n ) = f (m) (α) m! e n m [ + A e n + A e n + A e n + A 4 e 4 n + O(e 5 n)], () f (x n ) = f (m) (α) (m )! e n m [ + B e n + B e n + B e n + B 4 e 4 n + O(e 5 n)], (4) where A k = m! θ (m+k)! k, B k = (m )! θ (m+k )! k and θ k = f (m+k) (α) for k N. f (m) (α) Dividing () by (4), we have the following relations: f(x n) f (x n) = m [e n K e n K e n K e 4 n + O(e 5 n)], (5) where K = A + B, K = A + A B B + B and K = A + A B A B + B + A B B B + B. Letting t = γ/m with (5), we obtain y n = x n γ f(xn) f (x n) = α + te + K ( t)e n + K ( t)e n + K ( t)e 4 n + O(e 5 n). (6) Evaluating f(y n ) and f (y n ) with e n replaced by y n α in (6), we get the following relations:

4 74 Young Hee Geum f(y n ) = f (m) (α)e m m! {t m + t m (A t + K m( t))e n + (tm (K (m )m(t ) A K (m + )(t )t + t(a t + K m( t)))e n) + 6 (tm ( K m( m + m )(t ) + A K m(m + )(t ) t 6K (t )t(a (m + )t +K m(m )( t) + 6t (K m( t) + t(a K (m + )(t ) + A t )))e n) + O(e 4 n)}. (7) where f (y n ) = mt m e m β ( + t T e + t T e + 6 t T e + O(e 4 ) ), (8) { T = B t K (m )(t ), T = K (m )(m )(t ) B K m(t )t + t ( B t + K (m )(t ) ), T = K (m )(m )(m )(t ) + B K m(m )(t ) t 6K (t )t ( K (m )(m )(t ) + B ( + m)t ) + 6t ( K (m )( t) + t ( B K m(t ) + B t )). Substituting ()-(8) into (), we obtain the error equation: e n+ = y n α K f = ψ e n + ψ e n + ψ e n + ψ 4 e 4 n + O(e 5 n) (9) where K f = y n b F (yn) f (x n) c f(xn) f (y n) F (yn) d, ψ f (y n) = t+ t m ( ct(t+t m ρ) (dt+bt m )(t+t m (m( +t)λ+ρ))) m(t+t m ρ) and coefficients ψ i, (i =,, 4) depend on the parameters t, b, c, d, λ, ρ and the function f(x). Solving ψ =, ψ = for b and c, respectively, we obtain b = t( (c+d)t m (t+t m ρ)+m(t+dλ dtλ+t m ρ)) t+mt +m λ+t m ( mλ+ρ), () c = dω (t+t m (m(t )λ+ρ)) +mt m (ω +t +m ( m(t )( +m(t ) +t )λ+ρ)) ω (ω +t +m (m(t )λ+ρ)) () with ω = (t )( + m(t ) + t) and ω = t + t m ρ(m(t )λ + ρ). We substitute b, c into ψ and put ψ = ψ θ + ψ θ. Solving ψ = independently of θ and θ, i.e. solving ψ = ψ = for d and t, we obtain d = (t η t +m η t m η +t +m η 4) (t ) (+m(t )+t) λ(m(t )λ+ρ) t m, () η = ( (m )m (m )mt + 4 ( + m ) t ( + m) t ), η = ( t ( + t) + m (t )( + (t )t) + m ( t + 4t )) ρ(m(t )λ + ρ), η = ( t ( + t) + m (t )( + (t )t) + m ( t + 4t )) ρ (m(t )λ + ρ), η 4 = m 4 (t ) 6 λ + mt ( 5 + t ( 4t + t )) λ + m (t ) ( + t(6 + t( + (t )t)))λ m( t) ( + t)ρ 6t ( + t)ρ + m (t )(( + t)( + t(6 + t( + (t )t)))λ ( + (t )t)ρ),

5 Iterative methods for multiple roots and their dynamics 74 and Substituting t = following relations: m m+ t = m m+. () into (), () and () with κ = ( m +m )m, we get the b = m (4κρ m(+m)(+κρ)) 6mκλ 8(+m)κρ, (4) c = υ+κm(m+) (4κ λρ(m(λ ρ) ρ)+κ(λ ρ)(λm (m+)ρ)+mρ) 6κλ(mλ (+m)ρ), (5) with υ = m (( + m) 4(8 + m(5 + m))κλ) + ( + m) 5 κ ρ and d = υ m (+m) κ( +κλ)ρ m(+m) 4 κ ( +κλ)ρ 6κλ(mλ (+m)ρ). (6) By the aid of symbolic computation of Mathematica[], we arrive at the relation below: e n+ = ψ 4 e 4 n + O(e 5 n), (7) where 4 m+κρ+mκρ) ψ 4 = 8+4m +4m +6m 4 +m 5 +m(+ θ m 4 (+m) (+m) θ m(m+) (m+) θ + m θ (m+) (m+)(m+). As a result, the proof is completed. 4 Numerical examples In this section, we present the typical computational experiments of proposed methods and illustrate the complex dynamical behavior related to the basins of attraction[,,] of iterative maps. Throughout the experiments with Mathematica, the error bound is x n α < 8 and significant digits are assigned as the minimum number of precision digits, to achieve the desired accuracy ensuring convergence of the proposed methods. It is essential to compute e n = x n α with high accuracy for desired numerical results. When a root α is not exact, we increase the precision that it uses internally in order to get a more accurate value. As a first instance, we choose a test function f(x) = (e x +4 ) 4 (x + 4) with a zero α = i of multiplicity m = 7 and an initial approximation x =.97i. As a second experiment, we take another test function f(x) = (e x +π ) 4 (x +π) 4 (log(x +π +)) 4 cot(x +7) having a root α = π i of multiplicity m = and choosing an initial value x =.97i. Throughout these examples, we confirm that the order of convergence is four and the computed asymptotic error constant e n+ /e 4 n well approaches the theoretical value η. The order of convergence and the asymptotic error constant are clearly shown in Tables - reaching a good agreement with the theory developed in Section

6 74 Young Hee Geum Table : Convergence behavior with f(x) = (e x +4 ) 4 (x + 4) (m, λ, ρ) = (7,, 5/), α = i n x n f(x n) x n α e n+ /e 4 n η log(e n/η) log(e n ).97i i i i i Table : Convergence behavior with f(x) = (e x +π ) 4 (x + π) 4 (log(x + π + )) 4 cot(x + 7) (m, λ, ρ) = (,, 7 ), α = π i n x n f(x n) x n α e n+ /e 4 n η log(e n/η) log(e n ).97i i i i i i The proposed methods () have the efficiency index EI = 4 /.587 agreeing with Kung-Traubs optimality conjecture. If both α and x n have the same accuracy of $MinP recision =, then e n = x n α would be nearly zero as n becomes large. When we calculate e n+ /e 4 n, overflow causes unfavorably. Computed values of x n are accurate up to significant digits. To obtain the trustworthy convergence behavior, we desire α with enough accuracy of 6 digits higher than MinP recision, which has 8 significant digits. Even if the number of significant digits of x n and α are and 8, respectively, we present the two values at most up to 5 significant digits due to the limited paper space. From now on, we discuss the dynamical behavior of iterative maps (). Choosing an initial guess close to a zero α is one of the most important factors to confirm the convergence of of iterative function. However, it is complicate to consider how close the initial values are to a zero α. An easy way of employing stable initial values is to utilize visual basins of attraction. In view of inspection of the area of convergence on the basins of attraction, the larger area of convergence would imply a better method. The quantitative analysis is needed to measure the size of area of convergence. Finally, we show Tables 4-7 featuring statistical data for the average number of iterations per point and the

7 Iterative methods for multiple roots and their dynamics 74 number of divergent points including CPU time. In all the cases, the region of investigation is a 6 by 6 square region centered at the origin and including all the zeros of the test polynomial functions. A 6 6 uniform grid in the square is taken to display initial points for the iterative methods via basins of attraction. Each grid point of a square is colored in accordance with the iteration number for convergence and the root it converged to α. This way we are able to discover 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 () with the desired basins of attraction, we take various polynomials having multiple roots with multiplicity m =,, 5, 6. Statistical data for the basins of attraction are tabulated in Tables 4-7. In this tables, 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 4 z) whose roots z =,,.5 ± i are all with multiplicity m =. Based on Table 4 and Figure, we realize that Y and Y5 is best in view of AVG and TDIV. As can be seen in Figure, Y has shown considerable amount of black point. These points causing divergence behavior were expected from the last column of Table 4. The best result for CPU is by Y6 and the worst one is by Y. As our next sample, the polynomial P (z) = (z x+7) has the complex roots z =.5 ± i of multiplicity m =. The results are listed in Table 5 and Figure. The method Y, Y4 and Y5 perform best in view of AVG and TDIV. As can be seen in Figure, Y and Y6 have shown a few black points. The best result for CPU is by Y5 and the worst one is by Y. As the third example, we take the polynomial P (z) = (z z) 5 whose roots z =, are all real with multiplicity m = 5. The results are listed in Table 6 and Figure. The method Y4 is best in view of AVG and TDIV. The best result for CPU is by Y and the worst one is by Y. In the last example, we use the polynomial P 4 (z) = (z /) 6 whose roots z = ±.777 are all real with multiplicity m = 6. The results are presented in Table 7 and Figure 4. The method Y4 is best in view of AVG and TDIV. The best result for CPU is by Y4 and the worst one is by Y. Acknowledgements. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Project Number: NRF-5RDAA88).

8 744 Young Hee Geum References [] 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. de Carvalho, M. Teixeira, Basin of attraction of a cuspfold singularity in D piecewise smooth vector fileds, Journal of Mathematical Analysis and Applications, 48 (4), -. [] C. Chun, B. Neta, Basins of attraction for several optimal fourth order methods for multiple roots, Mathematics and Computers in Simulation, (4), [4] Y.H. Geum, Y.I. Kim, Cubic convergence of parameter-controlled Newton-secant method for multiple zeros, Journal of Computational and Applied Mathematics, (9), [5] 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. Article ID [6] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, Journal of the ACM, (974), [7] J. L. Morris, Computational Methods in Elementary Numerical Analysis, John Wiley & Sons, 98. [8] M.S. Petković, L.S. Petković, J. Džunić, Accelerating generators of iterative methods for finding multiple roots of nonlinear equations, Computers and Mathematics with Applications, 59 (), [9] 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, 964. [] S. Wolfram, The Mathematica Book, Wolfram Media,. Received: November, 5; Published: December, 5

9 Iterative methods for multiple roots and their dynamics 745 Table 4: Typical Example with P (z) = (z 4 z), m = METHOD CPU TCON AVG TDIV Y , 6.54 Y , Y Y , Y5 8. 6, 6.54 Y , Table 5: Typical Example with P (z) = (z x + 7), m = METHOD CPU TCON AVG TDIV Y , Y Y Y , Y , Y Table 6: Typical Example with P (z) = (z z) 5, m = 5 METHOD CPU TCON AVG TDIV Y , Y 6.9 6, Y 4.8 6, 5.65 Y , Y5.99 6, Y ,

10 746 Young Hee Geum Table 7: Typical Example with P 4 (z) = (z /) 6, m = 6 METHOD CPU TCON AVG TDIV Y , Y , 5.6 Y 4.5 6, Y4. 6, Y , Y , Figure : The top left for Y, the top center for Y, and the top right for Y, the buttom left for Y4, the buttom center for Y5 and the buttom right for Y6, for the roots of the polynomial P (z) = (z 4 z).

11 Iterative methods for multiple roots and their dynamics Figure : The top left for Y, the top center for Y, and the top right for Y, the buttom left for Y4, the buttom center for Y5 and the buttom right for Y6, for the roots of the polynomial P (z) = (z x + 7). Figure : The top left for Y, the top center for Y, and the top right for Y, the buttom left for Y4, the buttom center for Y5 and the buttom right for Y6, for the roots of the polynomial P (z) = (z z) 5.

12 748 Young Hee Geum Figure 4: The top left for Y, the top center for Y, and the top right for Y, the buttom left for Y4, the buttom center for Y5 and the buttom right for Y6, for the roots of the polynomial P 4 (z) = (z /) 6.