5. Numerical Solution of Large Linear Systems

This chapter is dedicated to a review of numerical methods to solve iteratively linear systems which, even with the most powerful computational platform today available, cannot be solved with direct methods. The chapter starts with the review of consistently ordered matrices and the related concept of the so-called “Property A”, which is a necessary condition for large linear systems to guarantee convergence of the related numerical schemes. The Young-Frankel theory of successive over-relaxations (SOR) is then revised in detail including the sub-cases of Gauss-Seidel and Jacobi methods. Several numerical examples are given for simple matrices with and without the Property A. A review of relaxation methods based on minimization of functional is given, and gradient methods are introduced as a subclass of variational methods.