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Higher order methods of the basic family of iterations via $$S$$-iteration scheme with $$s$$-convexity. (English) Zbl 1436.65060
Let $$p(z)=\sum_{i=0}^n}a_iz^$$ and $D_m(z)=\det \left ( \begin{array}{ccccc} p'(z) & \frac{p''(z)}{2!} & \ldots & \frac{p^{(m-1)}(z)}{(m-1)!} & \frac{p^{(m)}(z)}{m!}\\ p(z) & p'(z) & \ddots & \ddots & \frac{p^{(m-1)}(z)}{(m-1)!} \\ 0 & p(z) &\ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \frac{p''(z)}{2!} \\ 0 & 0 & \ldots & p(z) & p'(z) \end{array} \right ).$ It is well known that typical root finding methods are defined as: $B_m(z)=z-p(z)\frac{D_{m-2}(z)}{D_{m-1}(z)}.$
Here the authors propose the following modification of the iteration process by replacing the convex combination with an $$s$$-convex one: $\begin{array}{l} z_{n+1}=(1-\alpha )^sB_m(z_m)+\alpha ^sB_m(v_n) \\ v_n=(1-\beta)^sz_n+\beta^sB_m(z_n), \end{array}$ where $$\alpha \in (0,1],\: \beta \in (0,1],\: s \in (0,1]$$. The algorithm for the generation of polynomiograph is presented in Section 4. Some graphical and numerical examples are presented in Section 5.
##### MSC:
 65H04 Numerical computation of roots of polynomial equations 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
##### Keywords:
root finding; $$S$$-iteration; $$s$$-convexity; polynomiography
PLCP; SQPlab
Full Text:
##### References:
  Agarwal, Rp; O’Regan, D.; Sahu, Dr, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8, 1, 61-79 (2007) · Zbl 1134.47047  Ardelean, G., Comparison between iterative methods by using the basins of attraction, Appl. Math. Comput., 218, 1, 88-95 (2011) · Zbl 1252.65086  Ardelean, G.; Balog, L., A qualitative study of Agarwal et al. iteration procedure for fixed points approximation, Creat. Math. Inform., 25, 2, 135-139 (2016)  Ardelean, G.; Cosma, O.; Balog, L., A comparison of some fixed point iteration procedures by using the basins of attraction, Carpathian J. Math., 32, 3, 277-284 (2016) · Zbl 1399.65118  Bonnans, Jf; Gilbert, Jc; Lemaréchal, C.; Sagastizábal, Ca, Numerical Optimization: Theoretical and Practical Aspects (2006), Berlin: Springer, Berlin  Burden, Rl; Faires, Jd, Numerical Analysis (2011), Boston: Brooks/Cole, Boston  Cordero, A.; Torregrosa, Jr; Vassileva, Mp, Three-step iterative methods with optimal eight-order convergence, J. Comput. Appl. Math., 235, 10, 3189-3194 (2011) · Zbl 1215.65091  Ding, Y.; Sui, C.; Li, J., An experimental investigation into combustion fitting in a direct injection marine diesel engine, Appl. Sci., 8, 12, 2489 (2018)  Ferreira, Nc; Caramelo, Fj; De Lima, Jjp; Guerreiro, C.; Botelho, Mf; Costa, Dc; Araújo, H.; Crespo, P.; De Lima, Jjp, Imaging methodologies, Nuclear Medicine Physics, 209-334 (2011), Boca Raton: CRC Press, Boca Raton  Gdawiec, K., Fractal patterns from the dynamics of combined polynomial root finding methods, Nonlinear Dyn., 90, 4, 2457-2479 (2017) · Zbl 1393.28006  Gdawiec, K.; Kotarski, W., Polynomiography for the polynomial infinity norm via Kalantari’s formula and nonstandard iterations, Appl. Math. Comput., 307, 17-30 (2017) · Zbl 1411.37047  Gdawiec, K., Kotarski, W., Lisowska, A.: Polynomiography based on the non-standard Newton-like root finding methods. Abstr. Appl. Anal. 2015, Article ID 797594 (2015) · Zbl 1386.65135  Gościniak, I.; Gdawiec, K., Control of dynamics of the modified Newton-Raphson algorithm, Commun. Nonlinear Sci. Numer. Simul., 67, 76-99 (2019)  Ishikawa, S., Fixed points by a new iteration method, Proc. Am. Math. Soc., 44, 1, 147-150 (1974) · Zbl 0286.47036  Jin, Y.; Kalantari, B., A combinatorial construction of high order algorithms for finding polynomial roots of known multiplicity, Proc. Am. Math. Soc., 138, 6, 1897-1906 (2010) · Zbl 1286.05176  Kalantari, B., On the order of convergence of a determinantal family of root-finding methods, BIT Numer. Math., 39, 1, 96-109 (1999) · Zbl 0922.65037  Kalantari, B., Generalization of Taylor’s theorem and Newton’s method via a new family of determinantal interpolation formulas and its applications, J. Comput. Appl. Math., 126, 1-2, 287-318 (2000) · Zbl 0971.65040  Kalantari, B., Polynomiography and applications in art, education, and science, Comput. Graph., 28, 3, 417-430 (2004)  Kalantari, B., Polynomial Root-Finding and Polynomiography (2009), Singapore: World Scientific, Singapore · Zbl 1218.37003  Kalantari, B.; Gerlach, J., Newton’s method and generation of a determinantal family of iteration functions, J. Comput. Appl. Math., 116, 1, 195-200 (2000) · Zbl 0982.65065  Kang, Sm; Alsulami, Hh; Rafiq, A.; Shahid, Aa, $$S$$-iteration scheme and polynomiography, J. Nonlinear Sci. Appl., 8, 5, 617-627 (2015) · Zbl 1327.30011  Karakaya, V.; Doğan, K.; Atalan, Y.; Bouzara, Neh, The local and semilocal convergence analysis of new Newton-like iteration methods, Turk. J. Math., 42, 3, 735-751 (2018) · Zbl 1436.49038  Lotfi, T.; Sharifi, S.; Salimi, M.; Siegmund, S., A new class of three-point methods with optimal convergence order eight and its dynamics, Numer. Algorithms, 68, 2, 261-288 (2015) · Zbl 1309.65054  Mann, Wr, Mean value methods in iteration, Proc. Am. Math. Soc., 4, 3, 506-510 (1953) · Zbl 0050.11603  Noor, Ma, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 1, 217-229 (2000) · Zbl 0964.49007  Picard, E., Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, J. Math. Pures Appl., 6, 4, 145-210 (1890) · JFM 22.0357.02  Pinheiro, Mr, $$s$$-convexity—foundations for analysis, Differ. Geom. Dyn. Syst., 10, 257-262 (2008) · Zbl 1156.26303  Rafiq, A.; Tanveer, M.; Nazeer, W.; Kang, Sm, Polynomiography via modified Jungck, modified Jungck Mann and modified Jungck Ishikawa iteration scheme, PanAm. Math. J., 24, 4, 66-95 (2014) · Zbl 1319.47058  Sahu, Dr; Singh, Kk; Singh, Vk, Some Newton-like methods with sharper error estimates for solving operator equations in Banach spaces, Fixed Point Theory Appl., 2012, 78 (2012) · Zbl 1273.49033  Sahu, Dr; Yao, Jc; Singh, Vk; Kumar, S., Semilocal convergence analysis of S-iteration process of Newton-Kantorovich like in Banach spaces, J. Optim. Theory Appl., 172, 1, 102-127 (2017) · Zbl 1359.65089  Varona, Jl, Graphic and numerical comparison between iterative methods, Math. Intell., 24, 1, 37-46 (2002) · Zbl 1003.65046
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