ABSTRACT
Despite the fact that the two theories have been developed almost independently, there are several connections between numerical analysis and the theory of difference equations. In the previous chapters, some common linear problems have seldom been considered. In this chapter, we shall explore more deeply some of these connections. In Sections 6.1 to 6.3, iterative methods for solving nonlinear equations are discussed, and the importance of employing the theory of difference inequalities is emphasized. Sections 6.4 and 6.5 deal with certain classical algorithms, such as Clenshaw, Miller, etc., from the point of view of the theory of difference equations. Sections 6.6 and 6.7 are devoted to the study of monotone iterative techniques, which offer monotone sequences that converge to multiple solutions of nonlinear equations. This study also includes an extension of monotone iterative methods to nonlinear equations with a singular linear part as well as applications to numerical analysis. In Section 6.8 we provide related problems of interest.
