TABLE OF CONTENTS There are a number of techniques for converting a nonsymmetric linear system into a symmetric one. One such technique solves the equivalent linear system A TAx = A Tb, called the normal equations. Often, this approach is avoided in practice because the coefficient matrix A TA is much worse conditioned than A. However, the normal equations approach may be adequate in some situations. Indeed, there are even applications in which it is preferred to the usual Krylov subspace techniques. This chapter covers iterative methods that are either directly or implicitly related to the normal equations.
In order to solve the linear system Ax = b when A is nonsymmetric, we can solve the equivalent system
which is symmetric positive definite (SPD). This system is known as the system of the normal equations associated with the least-squares problem
Note that (8.1) is typically used to solve the least-squares problem (8.2) for overdetermined systems, i.e., when A is a rectangular matrix of size n m, m < n.
A similar well-known alternative sets x = A Tu and solves the following equation for u:
Once the solution u is computed, the original unknown x can be obtained by multiplying u by A T. However, most of the algorithms we will see do not invoke the u variable explicitly but work with...
