ABSTRACT

  In this dissertation, we construct iteration schemes for nonlinear problems in various types. The presented schemes are iterative methods for the scalar and system of nonlinear cases as well as for calculating matrix sign function.

  In the scalar case, we designed efficient iteration schemes with memory to solve nonlinear equations in the class of Steffensen–type methods which is derivative free and without the use of additional functional evaluations percomputing step.

  We briefly remind some of the well- known commands in Mathematica to calculate nonlinear equation, system of nonlinear equation and matrix sign function.

  Here the aim is to use the accelerating technique via interpolating polynomials obtained from the concept of with memorization. The computational efficiency index is improved for the schemes and convergence is confirmed by the study of dynamics of the proposed iterative methods via basins of attraction. Numerical investigations are given.

  For the matrix sign function, some iterative schemes are designed. Some cases of the free involve parameter have been shown the global convergence of the proposed method with high rate of convergence. It is discussed analytically the stability of the scheme. Finally, a variety of experiments is given to show the efficiency of the proposed method. The iterative methods for solving system of nonlinear equations are investigated also at the end of this dissertation. For this purpose, a higher-order multi-point iteration scheme is constructed. The convergence analysis is proved.

  In addition, the computational efficiency index will be thoroughly discussed alongside numerical comparisons with existing methods. Finally, we illustrate some numerical examples to support the theoretical results.