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Numerical Linear Algebra with Applications by William Ford

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Chapter 21

Krylov Subspace Methods

Abstract

This chapter presents iterative methods for the solution of large, sparse, systems. It begins by describing the CRS format for the representation of a sparse matrix. Following this is a detailed development of the conjugate gradient method. The approach is to first develop the method of steepest descent and then show how it can be improved, leading to the conjugate gradient algorithm (CG). A convergence result is stated but not proved. Preconditioning is introduced, and two methods for preconditioning CG are presented: incomplete Cholesky and SSOR. Examples are provided to illustrate the need for preconditioning CG. A Krylov subspace is defined, and it is shown that CG is actually a Krylov subspace ...

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