ABSTRACT
This chapter examines conditions for the monotone convergence of single-step methods generated by special point-to-point mappings and the monotone convergence of Newton-like iterations of the form. Iterations of this type are extremely important in the areas mentioned in practical applications. The chapter considers some fixed point theorems which hold in arbitrary complete lattices. These theorems are originally due to Tarski. By the theorem, the existence of a fixed point for every increasing function is a necessary condition for the completeness of a lattice. The chapter also examines the monotonicity of the single variable Newton method. In some other cases the convergence of these sequences can follow from conditions other than regularity. The chapter provides some examples that satisfy conditions and explains how the general results can be applied to obtain monotone convergence theorems for Newton's and secant methods.
