ABSTRACT
This chapter begins with considering the most general and important notions of the theory of numerical methods, which are equally applicable to all reasonable approximations of a given operator equation, including all types of grid methods. It considers the most essential questions in the theory of iterative methods with model operators, starting with an analysis of the modified method of the simple iteration for nonlinear systems regarded as operator equations in the Euclidean space H. The chapter emphasizes the attraction of the appropriate symmetrization of a linear system with a general invertible operator and examines the convergence estimates in more detail. It also considers some important and in many respects unresolved aspects of using nonsymmetric model operators. The goal is to analyze in detail what such a procedure requires of computational work to ensure that the desired accuracy is achieved.
