Abstract
The problem of solving equations and systems of nonlinear equations is among the most important in theory
and practice, not only of applied mathematics but also in many branches of science, engineering, physics, computer
science, astronomy, finance,. . . The large number of scientists who have recently worked on this topic show a high
level of contemporary interest. Although the rapid development of the digital computers led to the e®ective imple-
mentation of many numerical methods, in the practical realization, it is necessary to solve several problems such
as computational efficiency based on the time used by the processor, the design of iterative methods that possess
a rapid convergence to the desired solution, the control of rounding errors, the information on the limits of error of
the approximate solution obtained, indicating the verifiable initial conditions to guaranty a true convergence, and
so on. Such problems constitute the starting point of this memory.
The overall objective of this report lies in the search of new and efficient iterative methods for equations and
systems of nonlinear equations. The origin is the work done by Weerakoon and Fernando which developed in one-
dimensional variant of Newton's method that uses the trapezoidal quadrature formula, achieving convergence of
order three. ÄOzban extended this idea, and got some new methods with third order convergence. Moreover, these
methods are special cases of the family of variants of Newton's method of order three defined by M. Frontini and
E. Sormani, using a generic quadrature formula of equispaced nodes. So, first, based on the idea of M. Frontini and
E. Sormani, we use the Gaussian quadrature formula generated and developed in Chapter 3 to get a set of families
of iterative methods for equations. All the set methods are predictor-corrector type where the prediction is done
initially with the Newton method. We show that the order of convergence is three under certain conditions imposed
on orthogonal polynomials that define the corresponding Gaussian quadrature family and five depending on the
behavior in the solution of the second derivative of the function which defines the nonlinear equation. Seeking to
improve the order of convergence of all iterative methods developed, we modified the algorithms obtained using
other predictor methods that exceed the order two of Newton. Initially we used as a predictor Traub's method which
has order of convergence three, getting a group of families of fifth-order iterative methods (or eleven under certain
conditions imposed on orthogonal polynomials and the second derivative of the function defining the equation).
Subsequently, we use as predictors Ostrowski and Kou's schemes whose order of convergence is four, getting the
whole family of iterative methods of order 6 and 11 under the same conditions.
Following the trend in the development of new iterative methods in Chapter 4 we have been developed various
methods of high order of convergence and optimal efficiency index according to the Kung-Traub conjecture (this
conjecture, along with other basic concepts are developed in Chapter 2).
Chapter 5 is dedicated to finding new iterative methods for systems of nonlinear equations. Generally, to
increase the convergence order further functional evaluations of the Jacobian matrix and the nonlinear function are
required. In this sense, we use some of the methods developed for equations in the Chapter 3 adapted to systems,
while others are developed specifically for systems.
Chapter 6 is devoted to the relationship between iterative methods for solving nonlinear equations and initial
value problems. Quadrature formulas in general and in particular those of Gauss, allow us to determine numerically
the solutions of both equations or systems of nonlinear equations (generating iterative methods as described herein)
as well as initial value problems, linear or not. Thus, quadrature formulas are the link between both problems.
We end this report by presenting the most relevant conclusions obtained and propose some open problems that
will constitute our future research.