Iterative Methods for Solving Large Sparse Systems of Linear Algebraic Equations

Abstract

Publisher Summary This chapter discusses the iterative methods for solving large sparse systems of linear algebraic equations. The chapter introduces two model problems and also the necessary matrix preliminaries. Linear stationary iterative methods of the first degree are described, along with their corresponding convergence criteria. The particular methods detailed are the basic Jacobi, Gauss–Seidel, successive overrelaxation (SOR), and symmetric successive overrelaxation (SSOR) methods. Only successive overrelaxation has any practical claim for direct implementation. However, the Jacobi and SSOR methods are included because they can be accelerated substantially using Chebyshev or conjugate gradient acceleration procedures. The chapter also focuses on how the size of the system can possibly be reduced using a so-called “red-black” ordering on rectangular domains. Alternating-direction implicit (ADI) methods are represented by the Peaceman–Rachford method. A brief comparison of selected iterative methods is also provided in the chapter.