Relaxation Acceleration in Iterative Methods

Abstract

A way to rise efficiency of algorithms using the method of simple iteration to find an immovable point a with any basic algorithm A is proposed. The way is reduced from the principle of minimality of next iteration error estimation when the current iteration xk, its image A(xk) and the estimation ||A(xk) - α || ≤ c||xk - α||p, p ≥ 1 are known. Here α, xk and A(xk) are elements of Euclidean or Hilbert space. The principle leads to recurrent formula yk+l=Yk+γk (A(yk) - Yk), where γk is the best for exactness of yk+1 numerical coefficient. In all iterations determined estimations of errors for the modification are not worse than ones provided by the simple iteration (in one-dimensional space are always better). A real quality of modification appears at statistical view on the error estimation. Its mathematical expectation at reasonable probabilistic hypothesis is notably lower than the estimation for the method of simple iteration.