Elementary Linear Algebra, 11th Edition

CHAPTER 9Numerical Methods

CHAPTER CONTENTS

1. 9.1 LU-Decompositions
2. 9.2 The Power Method
3. 9.3 Comparison of Procedures for Solving Linear Systems
4. 9.4 Singular Value Decomposition
5. 9.5 Data Compression Using Singular Value Decomposition

INTRODUCTION

This chapter is concerned with “numerical methods” of linear algebra, an area of study that encompasses techniques for solving large-scale linear systems and for finding numerical approximations of various kinds. It is not our objective to discuss algorithms and technical issues in fine detail since there are many excellent books on the subject. Rather, we will be concerned with introducing some of the basic ideas and exploring two important contemporary applications that rely heavily on numerical ideas—singular value decomposition and data compression. A computing utility such as MATLAB, Mathematica, or Maple is recommended for Sections 9.2 to 9.5.

9.1 LU-Decompositions

Up to now, we have focused on two methods for solving linear systems, Gaussian elimination (reduction to row echelon form) and Gauss–Jordan elimination (reduction to reduced row echelon form). While these methods are fine for the small-scale problems in this text, they are not suitable for large-scale problems in which computer roundoff error, memory usage, and speed are concerns. In this section we will discuss a method for solving a linear system of $n$ equations ...