Estudio de la dinámica del método de Newton amortiguado

  1. Magreñán Ruiz, Ángel Alberto
Supervised by:
  1. José Manuel Gutiérrez Jiménez Director
  2. Natalia Romero Álvarez Director

Defence university: Universidad de La Rioja

Fecha de defensa: 04 September 2013

  1. Luis Javier Hernández Paricio Chair
  2. Sonia Busquier Sáez Secretary
  3. Miquel Grau Sánchez Committee member
  4. Miquel Noguera Batlle Committee member
  5. María Purificación Vindel Cañas Committee member
  1. Matemáticas y Computación

Type: Thesis

Institutional repository: lock_openOpen access Editor


The Doctoral Thesis defended lies on the border of two lines of mathematical research of great relevance, such as dynamical systems and the numerical solution of nonlinear equations by iterative processes. Specifically, we studied the iterative method known as damped Newton method, which is a modification of the classical Newton method. This method generates a sequence depending on a damping parameter, which in suitable conditions, converges to the desired solution. The thesis shows the importance of the damping parameter, not only in the convergence of the method but also in their dynamic properties. The thesis presents three different approaches. The first one is focused in the analysis of the real dynamics of the method, using among other techniques Feigenbaum diagrams and Lyapunov exponents which allow us to find strange behaviors (convergence to cycles, chaotical behaviour, etc.) for different values of the damping parameter. In the second approach, dedicated to the complex dynamics of the method, the main aim consist on distinguishing the basins of attraction associated to the solutions, which have in many cases an intricate fractal structure. We rely on the character of this fixed point, the attracting cycles and in the parameters planes associated to the iteration of the free critical points. Finally, we study the method applied to operators defined between Banach spaces, obtaining results on the local and semilocal convergence. This generalization allows us to find the solution of different problems such as systems of nonlinear equations, differential or integral equations, optimization problems, etc.