High performance multidimensional iterative processes for solving nonlinear equations

Abstract

In a large number of problems in applied mathematics, there is a need to solve nonlinear equations and systems, since many problems eventually are reduced to these. As the difficulty of the systems increases, obtaining the analytical solution becomes more complex. Furthermore, with the growth of computational tools, the dimensions of the problems to be solved have increased exponentially, making it more essential to obtain an approximation to the solution in a simple way that does not require significant time and computational cost. That is one of the reasons why iterative methods have increased their importance in recent years, as a multitude of schemes have been designed to converge rapidly to the solution and, in this way, to be able to solve problems that would be more arduous to solve using classical tools. This Doctoral Thesis focuses on the study and design of numerous iterative methods that improve classical schemes in terms of their order of convergence, accessibility, number of solutions obtained or applicability to problems with special characteristics, such as non-differentiability or multiplicity of roots. The procedures studied in this report range from a family of optimal multi-step methods for solving equations, to a parametric derivative-free family of weight function schemes, to which memory is introduced for solving nonlinear systems. Additional procedures are described in this report such as iterative schemes that obtain roots with different multiplicities for equations and methods that approximate roots simultaneously for equations as well as for systems, and for simple as well as for multiples roots. In addition, part of this report focuses on how to perform the dynamical analysis for iterative schemes with memory that solve systems of nonlinear equations, as well as this study is carried out for different known iterative procedures. This dynamical analysis allows us to visualise and analyse the possible behaviours of the iterative methods depending on the initial approximations. The results described above form part of this Doctoral Thesis to obtain the title of Doctor in Mathematics.