Linear System Solvers: Sparse Iterative Methods [chapter]

Henk A. Van der Vorst, Tony F. Chan
1997 ICASE/LaRC Interdisciplinary Series in Science and Engineering  
In this chapter we will present a n o verview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and they include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, LSQR and GMRES. We will sketch h o w these methods can be derived from simple basic iteration formulas, and how they are interrelated. Iterative s c hemes are usually considered as an alternative for the solution of linear
more » ... solution of linear sparse systems, like those arising in, e.g., nite element or nite di erence approximation of systems of partial di erential equations. The structure of the operators plays no explicit role in any o f t h e s e s c hemes, and the operator may be given even as a rule or a subroutine. Although these methods seem to be almost trivially parallellizable at rst glance, this is sometimes a point of concern because of the inner products involved. We will consider this point in some detail. Iterative methods are usually applied in combination with so-called preconditioning operators in order to further improve convergence properties. This aspect will receive more attention in a separate chapter in the same volume.
doi:10.1007/978-94-011-5412-3_4 fatcat:jvtjrswbyvblfbzw66jztlpdkq