CHAPTER 6 - Adaptive Chebyshev Acceleration

Abstract

This chapter focuses on adaptive Chebyshev acceleration. Chebyshev algorithms can be used without modification to solve certain singular problems; however, the algorithms should not be used when the basic iteration matrix G has an eigenvector deficiency. The requirements for convergence and the rate of convergence for the secondary iterative process differ from those of the main iterative process. When adaptive parameter estimation procedures are used, the total iteration process involves the successive application of different Chebyshev polynomial sequences. Successive polynomial generation can enhance the adaptive procedures. The adaptive parameter estimation procedure utilizes only the 2-norm of the pseudoresidual vector δ. CHEBY subroutine is designed for use as a software package to provide the required acceleration parameters and to provide an estimate of the iteration error for the Chebyshev polynomial method. The convergence rate of the Chebyshev acceleration method depends strongly on the values of μ1 and μN but only weakly on the size of the matrix problem.