Techniques for Accelerating Iterative Methods for the Solution of Mathematical Problems.

摘要

Mathematical problems can be solved numerically by deriving an iteration scheme that generates a sequence, the limit of which is the solution of the problem. However, quite often the generated sequence converges very slowly or even diverges. This study describes and compares methods designed to accelerate a slowly convergent sequence or to obtain the solution of the problem from a divergent sequence. The methods studied are Aitken's Delta-Squared method, Wynn's epsilon and modified epsilon methods, the minimal polynomial extrapolation method, the reduced rank extrapolation method, and Anderson's generalized secant algorithms. The derivation of these methods, as applied to both linear and nonlinear problems, are presented. In addition, the acceleration methods are compared theoretically and numerically. Results are presented from extensive numerical testing of problems in an m-dimensional vector space. (kr)