http://scgroup.hpclab.ceid.upatras.gr/class/SCII/Various/SaadvdVorst2000.pdf

Iterative solution of linear systems in the 20th century

Some aspects of the parallel and distributed iterative algorithms—a survey

Theoremes de points fixes dans les treillis complets, etude du comportement d'un systeme dynamique discret, analyse semantique exacte des programmes et applications. Methodes constructives d'approximation de points fixes d'operateurs monotone sur un treillis complet. Analyse semantique approchee des programmes et applications. Analyse semantique des procedures recursives.

Méthodes itératives de construction et d'approximation de points fixes d'opérateurs monotones sur un treillis, analyse sémantique des programmes

Nous presentons dans cet article des resultats de convergence des algorithmes asynchrones bases essentiellement sur la notion classique de contraction.

Nous generalisons, en particulier, tous les resultats de convergence de ces algorithmes qui font l'hypothese de contraction en norme vectorielle qui recemment a ete tres souvant utilisee.

Par ailleurs, l'hypothese de contraction en norme vectorielle peut se trouver difficile, voire impossible a verifier pour certains problemes qui peuvent etre cependant abordes dans le cadre de la contraction classique que nous adoptons.

In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction.

We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently.

Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.

Some convergence results for asynchronous algorithms

Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.

https://www.math.temple.edu/~szyld/papers/Lanzkron.Rose.Szyld.91.pdf

Convergence of nested classical iterative methods for linear systems

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J. C. Miellou

Papers

Algorithmes de relaxation chaotique à retards