ABSTRACT
This chapter examines conditions for the convergence of special single-step methods generated by point-to-point mappings. It provides sufficient conditions for the convergence of the Newton method using the majorant theory. The speed of convergence is also examined using the theory of majorants. However, based on the special structure of the method more advanced convergence conditions can be derived, and better error estimates can be presented than those obtained from the general theory. The chapter also provides sufficient conditions for the convergence of iterations of the form. The discussion of L. V. Kantorovich’s analysis for multipoint iterative methods is less developed, however, although the fundamental theory of multipoint iterative methods was developed by A. M. Ostrowski and J. R. Traub in the early 1960s. A generalized norm is used which is defined to be a map from a linear space into a partially ordered Banach space. The metric properties of the problem can be better analyzed by this tool.
