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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)
Software:
PLCP; SQPlab
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