Iterative Methods for Linear Systems

This chapter discusses the iterative methods for solving systems of linear equations. The iterative methods are suited for problems involving large sparse matrices, much more so in most cases than direct methods such as Gaussian elimination. The chapter discusses the theory of nonnegative matrices. The theory of M-matrices has played a fundamental role in the development of efficient methods for establishing the convergence and in accelerating the convergence of the iterative methods. The chapter presents the formulas for the classical Jacobi and Gauss–Seidel iterative methods. The convergence results are related to two common classes of situations: (1) those in which A is a nonsingular M-matrix and second and (2) those in which A is hermitian and/or positive definite. A slight modification of the Gauss–Seidel method by the use of a relaxation can be used to produce a method, known as the successive over-relaxation (SOR) method, which can dramatically reduce the number of iterations in certain cases.