Dinàmica Holomorfa
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TFG-TFMMetadades
Títol
Dinàmica HolomorfaAutoria
Tutor/Supervisor; Universitat.Departament
Vindel Cañas, María Purificación; Universitat Jaume I. Departament de MatemàtiquesData de publicació
2022-07-18Editor
Universitat Jaume IResum
Numerical Methods are mathematical tools constructed in many scientific applications for solving a
mathematical problem which do not necessarily have an analytical solution. When one constructs a
numerical method, ... [+]
Numerical Methods are mathematical tools constructed in many scientific applications for solving a
mathematical problem which do not necessarily have an analytical solution. When one constructs a
numerical method, it is done with one obvious goal in mind: This method must behave properly, and
solve the problem in hand.
However, this is not always the case. Often, numerical methods either get caught in infinite loops,
and thus are unable to decide a definite answer, or behave chaotically altogether.
In this work we will study the behavior of holomorphic maps in the complex plane . At first we will
focus on simple polynomial functions, slowly moving towards the understanding of the behavior of
rational maps, such as the ones derived from the Newton-Raphson Method, when under iteration,
and the geometric sets and shapes that they produce.
The dynamics of this work will be as follows: We will present an exercise, work it out for a few values,
and then show what happens to all points in a specified region of the complex plane.
In order to properly understand the behavior of these functions, we will show a series of pictures and
visualizations.
It is important to note that the entirety of these images have been generated by using the software
Sempiternum, a visualizer of holomorphic dynamics.
The main sections of this work will analyze and explain the behavior of several holomorphic maps.
Moreover, there is an addendum at the end explaining the overall functionality of the Sempiternum
Software, as well as the main and most important sections of the code. [-]
Paraules clau / Matèries
Descripció
Treball Final de Grau en Matemàtica Computacional. Codi: MT1054. Curs: 2021/2022
Tipus de document
info:eu-repo/semantics/bachelorThesisDrets d'accés
info:eu-repo/semantics/openAccess