Time-lag Derivative Convergence for Fixed Point Iterations

Time-lag in Derivative Convergence for Fixed Point Iterations Le retard en convergence des dérivées pour les calculs itératifs avec point fixe

RÉSUMÉ. Une étude antérieure a prouvé et vérifié expérimentalement sur un code Euler 2D que les calculs itératifs avec point fixe peuvent être différentiés pour obtenir les dérivées aux premier et deuxième ordres des fonctions implicites définies par des équations d’état. On considérait également que les itérées correspondantes des gradients et Hessiens réduits convergent à la même vitesse que ...

Fixed Point Iterations

Recall that a vector norm on R is a mapping ‖·‖ : R → R satisfying the following conditions: • ‖x‖ > 0 for x 6= 0. • ‖λx‖ = |λ|‖x‖ for x ∈ R and λ ∈ R. • ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ R. Since the space Rn×n of all matrices is also a vector space, it is also possible to consider norms there. In contrast to usual vectors, it is, however, also possible to multiply matrices (that is, the matric...

Anderson Acceleration for Fixed-Point Iterations

This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Machinery, 12 (1965), 547-560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anders...

Fixed Point Iterations for Real Functions

On The Convergence Of Fixed Slope Iterations

We provide a local as well as a semilocal convergence analysis for a certain class of fixed slope iterations in a Banach space setting. Using a weaker Hölder condition on the operator involved and more precise estimates than in [1], [2] we provide in the semilocal case: finer error estimates on the distances involved and an at least as precise information on the location of the solution; in the...