Linear Algebraic Equations

Abstract

The solution of linear algebraic equations is probably the most important topic in numerical methods. Since the simplest models for the physical world are linear, linear equations arise frequently in physical problems. Even the most complicated situations are frequently approximated by a linear model as a first step. Further, as will be seen in Chapter 7, the solution of a system of nonlinear equations is achieved by an iterative procedure involving the solution of a series of linear systems, each of them approximating the nonlinear equations. Similarly, the solution of differential and integral equations using finite difference method leads to a system of linear or nonlinear equations. Linear equations also arise frequently in numerical analysis. For example, the method of undetermined coefficients which is useful for deriving formulae for numerical differentiation, integration or solution of differential equations, generally leads to a system of linear equations.