A family of simultaneous zero-finding methods

AbstractApplying Hansen-Patrick's formula for solving the single equation f(z) = 0 to a suitable function appearing in the classical Weierstrass' method, two one-parameter families of interation functions for the simultaneous approximation of all simple and multiple zeros of a polynomial are derived. It is shown that all the methods of these families have fourth-order of convergence. Some computational aspects of the proposed methods and numerical examples are given

Laguerre-like methods for the simultaneous approximation of polynomial multiple zeros

Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples