TABLE OF CONTENTS There are two types of practical methods for solving the equation (1.1): the direct methods (Chapter 2) and the iterative methods (Chapter 3). Each method produces a numerical solution x, that should approximate the analytical solution x*= A ?1 b with a certain number of accurate digits. Modern developments into new solution techniques make the distinction of the two types a bit blurred, because often the two are very much mixed in formulation.
Traditionally, a direct method refers to any method that seeks a solution to (1.1) by simplifying A explicitly
| (1.3) |  |
where T is a much simplified matrix (e.g. T is ideally diagonal) and c ? 
n. The philosophy is essentially decoupling the interactions of components of x = [ x 1, , x n] T in a new system. One may say that the enemy is A. A somewhat different approach is taken in the Gauss Purcell method Section 15.5 that views x from a higher space n +1.
On the other hand, without modifying entries of A, an iterative method finds a sequence of solutions x 0, x 1, , x k, by working closely with the residual vector
| (1.4) |  |
which can not only indicate how good x k is, but also may extract analytical information of matrix A. One hopes that an early termination...
