Numerical iterative methods for nonlinear problems.

The primary focus of research in this thesis is to address the construction of iterative methods for nonlinear problems coming from different disciplines. The present manuscript sheds light on the development of iterative schemes for scalar nonlinear equations, for computing the generalized inverse of a matrix, for general classes of systems of nonlinear equations and specific systems of nonlinear equations associated with ordinary and partial differential equations. Our treatment of the considered iterative schemes consists of two parts: in the first called the ’construction part’ we define the solution method; in the second part we establish the proof of local convergence and we derive convergence-order, by using symbolic algebra tools. The quantitative measure in terms of floating-point operations and the quality of the computed solution, when real nonlinear problems are considered, provide the efficiency comparison among the proposed and the existing iterative schemes. In the case of systems of nonlinear equations, the multi-step extensions are formed in such a way that very economical iterative methods are provided, from a computational viewpoint. Especially in the multi-step versions of an iterative method for systems of nonlinear equations, the Jacobians inverses are avoided which make the iterative process computationally very fast. When considering special systems of nonlinear equations associated with ordinary and partial differential equations, we can use higher-order Frechet derivatives thanks to the special type of nonlinearity: from a computational viewpoint such an approach has to be avoided in the case of general systems of nonlinear equations due to the high computational cost. Aside from nonlinear equations, an efficient matrix iteration method is developed and implemented for the calculation of weighted Moore-Penrose inverse. Finally, a variety of nonlinear problems have been numerically tested in order to show the correctness and the computational efficiency of our developed iterative algorithms