CHAPTER 7 - Conjugate Gradient Acceleration

Abstract

This chapter focuses on conjugate gradient (CG) acceleration. The CG method, though an iterative method, converges to the true solution of the linear system in a finite number of iterations in the absence of rounding errors. The CG method is a whole family of methods. Every such method can be regarded as an acceleration process for a particular linear stationary iterative method of first degree. The CG method can be represented in a three-term form, which resembles Chebyshev acceleration applied to the RF method. Conjugate gradient acceleration of a given iterative method converges with respect to a certain error measurement procedure at least as fast as the corresponding Chebyshev procedure. No parameter estimates are required in the implementation of CG acceleration. The chapter describes the three-term form of the CG method and the procedures for deciding when to terminate the iterative process. It highlights computational algorithms for carrying out the CG acceleration procedure.