Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices. These algorithms are the classical onesided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one-sided Jacobi to compute high accurate singular value decompositions of matrices given as rank-revealing factorizations. When ...

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds ...

The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on invariant subspace variations that are proportional to the reciprocals of absolute gaps between subsets of spectra or subsets of singular values. These bounds may be bad news for invariant subspaces corresponding to clustered eigenvalues or clustered singular values of much smaller m...

∗ ONERA, DPRS-SAGP, BP 72 29 avenue de la Division Leclerc, F-92322 Châtillon Cedex, France (e-mail: [email protected]) ∗∗ I3S-UNSA-CNRS, Les Algorithmes, Bât. Euclide B, 2000 Route des Lucioles, BP 121 06903 Sophia Antipolis Cedex, France (e-mail: [email protected]) ∗∗∗ ONERA, DPRS-SAGP, BP 72 29 avenue de la Division Leclerc, F-92322 Châtillon Cedex, France (e-mail: helene.piet-laha...

We study perturbation theory in certain quantum mechanics problems in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of the unusual techniques which are required to obtain perturbative expansions of the energies in such cases. These include a point-splitting prescription for exp...

Abstract. In this paper, we consider perturbations to a sequence of moments associated with an orthogonality linear functional that is represented by a positive measure supported in [−1, 1]. In particular, given a perturbation to such a measure on the real line, we analyze the perturbation obtained on the corresponding measure on the unit circle, when both measures are related through the Szeg ́́...

Singular perturbation methods are applied to an analysis of the operation of an isothermal gas step slider bearing of narrow geometry and operating at moderate bearing numbers. Approximate expressions are obtained for the pressure field in the lubricating gap, as well as the load-carrying capacity of the bearing; and the influence of the nature of the bearing step on those quantities is investi...

We consider slow–fast systems of differential equations, in which both the slow and fast variables are perturbed by additive noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the re...

The presence of fast and slow modes in vehicle suspension systems, based on a half car model, is utilized in the design of active suspension control using singular perturbation theory. This strategy is based on the slow-fast control design. The suspension system performance is optimised with respect to ride comfort, road holding and suspension rattle space as expressed by the mean-square-values...

In this article we deal with singularly perturbed Filippov systems Zε: