Beny Neta

Basin attractors for various methods

Abstract There are many methods for the solution of a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several methods of various orders and present the basin of attraction for several examples. It can be seen that not all higher order methods were created equal. Newton’s, Halley’s, Murakami’s and Neta–Johnson’s methods are consistently better than the others. In two of the examples Neta’s 16th order scheme was also as good.

On optimal fourth-order iterative methods free from second derivative and their dynamics

In this paper new fourth order optimal root-finding methods for solving nonlinear equations are proposed. The classical Jarratt’s family of fourth-order methods are obtained as special cases. We then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three. The basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance.

Two-step fourth-order P-stable methods with phase-lag of order six for y ″=( t,y )

Abstract We present a new family of two-step fourth-order methods which when applied to the test equation: y″ = − λ2y, λ>0, are at once P-stable and have a phase-lag of order H6 (H = λh, h is the step-size).

Basins of attraction for several methods to find simple roots of nonlinear equations

There are many methods for solving a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed.

High-order nonlinear solver for multiple roots

A method of order four for finding multiple zeros of nonlinear functions is developed. The method is based on Jarratts fifth-order method (for simple roots) and it requires one evaluation of the function and three evaluations of the derivative. The informational efficiency of the method is the same as previously developed schemes of lower order. For the special case of double root, we found a family of fourth-order methods requiring one less derivative. Thus this family is more efficient than all others. All these methods require the knowledge of the multiplicity.

Basin attractors for various methods for multiple roots

Abstract There are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss all known methods of orders two to four and present the basin of attraction for several examples.

A sixth-order family of methods for nonlinear equations

A one-parameter family of sixth-order methods for finding simple zeros of nonlinear functions is developed. Each member of the family requires three evaluations of the given function and only one evaluation of the derivative per step.

Some fourth-order nonlinear solvers with closed formulae for multiple roots

In this paper, we present six new fourth-order methods with closed formulae for finding multiple roots of nonlinear equations. The first four of them require one-function and three-derivative evaluation per iteration. The last two require one-function and two-derivative evaluation per iteration. Several numerical examples are given to show the performance of the presented methods compared with some known methods.

On a family of multipoint methods for non-linear equations

A new one-parameter family of methods for finding simple zeros of non-linear functions is developed. Each member of the family requires four evaluations of the given function and only one evaluation of the derivative per step. The order of the method is 16.

A third-order modification of Newton's method for multiple roots

In this paper, we present a new third-order modification of Newtons method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots.