Analysis of round-off errors

Periodicity and Transport from Round-Off Errors

AMS Subject Classi cation: 58F20, 60J60, 65G05 We investigate the effects of round-off errors on quasi-periodic motions in a linear symplectic planar map. By discretizing coordinates uniformly we transform this map into a permutation of Z2 , and study motions near infinity, which correspond to a fine discretization. We provide numerical evidence that all orbits are periodic and that the average...

Estimation of Round-Off Errors on Several Computer Architectures

Numerical validation of computed results in scienti c computation is always an essential problem as well on sequential architecture as on parallel architecture. The probabilistic approach is the only one that allows to estimate the round-o error propagation of the oating point arithmetic on computers. We begin by recalling the basics of the CESTAC method (Contrôle et Estimation STochastique des...

Estimation of Round-off Errors in OpenMP Codes

A KAM type theorem for systems with round-off errors

Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth periodic on the angle perturbations of the twist map, bounding the discretized map from above and from below. Using the well known Moser’s theorem we prove t...

Reducing round-off errors in symmetric multistep methods

Certain symmetric linear multistep methods have an excellent long-time behavior when applied to second order Hamiltonian systems with or without constraints. For high accuracy computations round-off can be the dominating source of errors. This article shows how symmetric multistep methods should be implemented, so that round-off errors are minimized and propagate like a random walk.