### A Method for Finding Sharp Error Bounds for Newton's Method Under the Kantorovich Assumptions.

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This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing...

This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the decay...

This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the...

We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions...

The error autocorrection effect means that in a calculation all the intermediate errors compensate each other, so the final result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the effect which appears in rational approximations to functions. The error autocorrection effect occurs...

We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete...

This article is a case study in the implementation of a portable, proven and efficient correctly rounded elementary function in double-precision. We describe the methodology used to achieve these goals in the crlibm library. There are two novel aspects to this approach. The first is the proof framework, and in general the techniques used to balance performance and provability. The second is the introduction of processor-specific optimization to get performance equivalent to the best current...

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations. Using more precise majorant conditions than in earlier studies, we provide: a larger radius of convergence; tighter error estimates on the distances involved; and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.

For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure...

We discuss the effects that the usual set theoretic and arithmetic operations with fuzzy sets and fuzzy numbers have with respect to the energies and entropies of the fuzzy sets connected and of the resulting fuzzy sets, and we also compare the entropies and energies of the results of several of those operations.

Nikdy neztotožnujme realitu s jejím matematickým či numerickým modelem. (Věnováno Emilu Vitáskovi k jeho 80. narozeninám.)