Masahide Kashiwagi

Verified Computations for Hyperbolic 3-Manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

A Urabe type convergence theorem for a constructive simplified Newton method in infinite dimensional spaces

A constructive simplified Newton method is presented for calculating solutions of infinite dimensional nonlinear equations, which uses a projection scheme from an infinite dimensional space to finite dimensional subspaces. A convergence theorem of the method is shown based on Urabes theorem. For a class of successive approximation methods containing an infinite dimensional homotopy method, a stopping criterion is shown using the convergence theorem. The authors show that under a certain condition the stopping criterion is satisfied in finite cycles of iteration.<<ETX>>

On Range Evaluation of Polynomials by Applying Interval Arithmetic

Interval arithmetic is able to be applied in the case that we evaluate the ranges of polynomials. When we evaluate the ranges of polynomials by applying the interval arithmetic, the problem that interval widths of the ranges increase extremely exists. Horner’s method is widely known as the evaluating method which mitigates this problem. The purpose of this paper is to propose the new methods which are able to mitigate this problem more efficiently than the Horner’s method. And in this paper, we show and com pare the efficiencies of the each new method by the results of some numerical examples.