Iterative Methods for Solving Systems of Equations

Abstract

Iterative Methods for Solving Systems of Equations It is well known that a wide class of problems, which arises in pure and applied sciences can be studied in the unified frame work of the system of absolute value equations of the type Ax − x = b, A ∈ Rn×n , b ∈ R n . Here x is the vector in R n with absolute values of components of x. In this thesis, several iterative methods including the minimization technique, residual method and homotopy perturbation method are suggested and analyzed. Convergence analysis of these new iterative methods is considered under suitable conditions. Several special cases are discussed. Numerical examples are given to illustrate the implementation and efficiency of these methods. Comparison with other methods shows that these new methods perform better. A new class of complementarity problems, known as absolute complementarity problem is introduced and investigated. Existence of a unique solution of the absolute complementarity problem is proved. A generalized AOR method is proposed. The convergence of GAOR method is studied. It is shown that the absolute complementarity problem includes system of absolute value equations and related optimizations as special cases