An Introduction to Numerical Methods and Analysis, 2nd Edition

CHAPTER 7

NUMERICAL METHODS FOR THE SOLUTION OF SYSTEMS OF EQUATIONS

Here we will look at numerical methods for solving two important problems from linear algebra:

1. The linear systems problem: Given a matrix A and a vector b, both known, find the vector x such that

2. The nonlinear systems problem: Given a vector-valued function F, find the vector x such that

A third important problem—the algebraic eigenvalue problem—is deferred to Chapter 8. These two chapters—7 and 8—are most heavily affected by the use of MATLAB, which in many ways was originally designed to be an easy-to-use interface for the FORTRAN packages LINPACK (linear systems) and EISPACK (eigenvalue problems). It is fair to ask why we are going to spend time describing in detail algorithms that can be executed with a single line of MATLAB code. Part of the answer lies in a bit of philosophy: The author believes very strongly that students need some exposure to the details of an algorithm in order to understand the material, but we will not go into deep detail on some of the more complicated algorithms, relying on the appropriate MATLAB constructs.

We begin with a review of the linear algebra concepts and notation that are especially necessary for this chapter.

7.1 LINEAR ALGEBRA REVIEW

A vector x is defined to be ...