Juan R. Torregrosa

Variants of Newton’s Method using fifth-order quadrature formulas

Abstract Some variants of Newton’s method are developed in this work in order to solve nonlinear equations depending on one or several variables, based in rules of quadrature of fifth order. We prove the third or fifth order of convergence of these methods for dimension one, and the second or third order in several variables, depending on the behaviour of the second derivative. Moreover, different numeric tests confirm or improve theoretic results and allow us to compare these variants with Newton’s classical method.

A modified Newton-Jarratt’s composition

A reduced composition technique has been used on Newton and Jarratt’s methods in order to obtain an optimal relation between convergence order, functional evaluations and number of operations. Following this aim, a family of methods is obtained whose efficiency indices are proved to be better for systems of nonlinear equations.

Chaos in King’s iterative family

In this paper, the dynamics of King’s family of iterative schemes for solving nonlinear equations is studied. The parameter spaces are presented, showing the complexity of the family. The analysis of the parameter space allows us to find elements of the family that have bad convergence properties, and also other ones with stable behavior.

Drawing Dynamical and Parameters Planes of Iterative Families and Methods

The complex dynamical analysis of the parametric fourth-order Kims iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).

Complex dynamics of derivative-free methods for nonlinear equations

The dynamical behavior of two iterative derivative-free schemes, Steffensen and M4 methods, is studied in case of quadratic and cubic polynomials. The parameter plane is analyzed for both procedures on quadratic polynomials. Different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or complex coefficients. The property of immersion of the basins of attraction in all cases is analyzed.

Dynamics of a family of Chebyshev-Halley type methods

In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This set has interesting similarities with the Mandelbrot set. The parameter space has allowed us to find different elements of the family which have bad convergence properties, since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method.

Variants of Newton's method for functions of several variables

Some variants of Newtons Method are developed in this work in order to solve systems of nonlinear equations, based in trapezoidal and midpoint rules of quadrature. We prove the quadratic convergence of one of these methods. Moreover, different numeric tests confirm theoretic results and allow us to compare these variants with Newtons classical method.

Three-step iterative methods with optimal eighth-order convergence

In this paper, based on Ostrowskis method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traubs conjecture. Numerical comparisons are made to show the performance of the new family.

Dynamics of a family of Chebyshev-Halley

In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This set has interesting similarities with the Mandelbrot set. The parameter space has allowed us to find different elements of the family which have bad convergence properties, since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method.

New modifications of Potra-Pták's method with optimal fourth and eighth orders of convergence

In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Ptaks method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traubs conjecture Kung and Traub (1974) [2], that establishes for an iterative method based on n evaluations an optimal order p=2^n^-^1 is fulfilled, getting the highest efficiency indices for orders p=4 and p=8, which are 1.587 and 1.682. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Ptaks method from which they have been derived, and with other recently published eighth-order methods.