Linear Systems of Equations

Many problems in mathematics lead to linear systems of equations. In fact, in using computers to solve such problems, we frequently encounter very large linear systems. Thus, the development of efficient algorithms to solve such systems is of central importance in numerical analysis. We differentiate between two types of methods. Direct methods solve the problem in a finite number of steps, and so are not subject to method error, although, of course, the results can be very badly affected by roundoff error. Indirect methods seek to find the solution by iteration, and thus usually lead only to approximate solutions since the iteration has to be stopped at some point. Although in this case we have both method and roundoff errors, iterative methods have their advantages. In this chapter we will primarily discuss direct methods. Iterative methods for linear system of equations will be discussed in Chapter 8.