Numerical Linear Algebra with Applications

Chapter 16

Least-Squares Problems

Abstract

This chapter presents linear least-squares. The normal equations are motivated by a figure showing that b − Ax should be normal to Ax. It is then shown that if m ≥ n, there is a unique least-squares solution if and only if x satisfies the normal equations. Also, x is unique if and only if A has full rank. The pseudoinverse or the Moore-Penrose generalized inverse is presented, and the condition number of an m × n matrix m ≥ n is defined using the pseudoinverse. There are three basic techniques for solving the overdetermined least-squares problem, m ≥ n, solving the normal equations, using the reduced QR decomposition, and using the reduced SVD. The most commonly used is the QR decomposition. The ...