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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 $$p(z)={\displaystyle \sum_{i=0}^n}a_iz^i$$ 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:
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