Chapter 1 - SAMPLE PROBLEMS

Abstract

This chapter discusses several mathematical problems for which the numerical solution requires solving a system of nonlinear equations. These problems are intended to give some idea of typical areas of numerical analysis where nonlinear systems of equations arise. A wide variety of problems in many areas, such as trajectory calculation or the study of oscillatory systems, may be stated in terms of a boundary value problem for an ordinary differential equation. Whenever a differential equation is approximated by an analogous difference equation, there arises the problem of estimating the discretization error. The chapter also discusses elliptic boundary value problems. Discretization of not only differential equations but also of other types of operator equations, such as integral equations or integrodifferential equations, leads to systems of equations in n dimensions. One of the most ubiquitous minimization problems is that of least-squares approximation. This problem typically arises in the course of attempting to estimate certain parameters in a functional relationship by means of experimental data.