Report Number: CS-TR-72-304
Institution: Stanford University, Department of Computer Science
Title: Richardson's non-stationary matrix iterative procedure.
Author: Anderssen, Robert S.
Author: Golub, Gene H.
Date: August 1972
Abstract: Because of its simplicity, Richardson's non-stationary iterative scheme is a potentially powerful method for the solution of (linear) operator equations. However, its general application has more or less been blocked by (a) the problem of constructing polynomials, which deviate least from zero on the spectrum of the given operator, and which are required for the determination of the iteration parameters of the non-stationary method, and (b) the instability of this scheme with respect to rounding error effects. Recently, these difficulties were examined in two Russian papers. In the first, Lebedev [1969] constructed polynomials which deviate least from zero on a set of subintervals of the real axis which contains the spectrum of the given operator. In the second, Lebedev and Finogenov [1971] gave an ordering for the iteration parameters of the non-stationary Richardson scheme which makes it a stable numerical process. Translation of these two papers appear as Appendices 1 and 2, respectively, in this report. The body of the report represents an examination of the properties of Richardson's non-stationary scheme and the pertinence of the two mentioned papers along with the results of numerical experimentation testing the actual implementation of the procedures given in them.