Family Of Root-finding Methods Research Articles

Complex dynamics of a sixth and seventh order family of root finding methods

This paper intends to investigate the dynamical behavior of a parameter based sixth and seventh order convergent family of iterative methods in the complex plane. The fixed points of the rational function obtained from this family are described with their stability. Further to this, the stability regions for the fixed points are displayed using graphical tools. Parameter planes for independent free critical points are presented to determine stable and poor members of the family. Dynamical planes are also provided for various iterative methods of the family having important numerical properties.

An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

Some variants of Hansen–Patrick method with third and fourth order convergence

Based on Hansen–Patrick method [E. Hansen, M. Patrick, A family of root finding methods, Numer. Math. 27 (1977) 257–269], we derive a two-parameter family of methods for solving nonlinear equations. All the methods of the family have third-order convergence, except one which has the fourth-order convergence. In terms of computational cost, all these methods require evaluations of one function, one first derivative and one second derivative per iteration. Numerical examples are given to support that the methods thus obtained are competitive with other robust methods of similar kind. Moreover, it is shown by way of illustration that the present methods, particularly fourth-order method, are very effective in high precision computations.

A family of root-finding methods with accelerated convergence

A parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given.

A family of root finding methods

A one parameter family of iteration functions for finding roots is derived. The family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newton's method. All the m...