Milestones in the Development of Iterative Solution Methods

Abstract

Iterative solution methods to solve linear systems of equations were originally formulated as basic iteration methods of defect-correction type, commonly referred to as Richardson's iteration method. These methods developed further into various versions of splitting methods, including the successive overrelaxation (SOR) method. Later, immensely important developments included convergence acceleration methods, such as the Chebyshev and conjugate gradient iteration methods and preconditioning methods of various forms. A major strive has been to find methods with a total computational complexity of optimal order, that is, proportional to the degrees of freedom involved in the equation. Methods that have turned out to have been particularly important for the further developments of linear equation solvers are surveyed. Some of them are presented in greater detail.