September 2014
Intermediate to advanced
628 pages
24h 57m
English
Content preview from Numerical Linear Algebra with Applications
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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 ...
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