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Akitoshi Takayasu,Shin’ichi Oishi,Takayuki Kubo 대한전자공학회 2009 ITC-CSCC :International Technical Conference on Ci Vol.2009 No.7
We propose a numerical verification method which proves the existence of solutions to two-point boundary value problems and leads their guaranteed error estimates for approximate solutions by finite element method. The ‘guaranteed’ error bound is rigorous, i.e. it takes every error such as the discretization error and the rounding error when solving the problems into account. We define an approximate solution operator for a linearized problem by some matrix form. The fixed-point formulation is led by operator equations. By using Banach’s fixed-point theorem, the existence of solution and guaranteed error bounds can be obtained. Finally, numerical example is presented.
Fast Filter for Verified Convex Hull and its Performance
Katsuhisa Ozaki,Takeshi Ogita,Shin’ichi Oishi 대한전자공학회 2009 ITC-CSCC :International Technical Conference on Ci Vol.2009 No.7
This paper is concerned with accurate algorithms in computational geometry. We propose fast algorithms to obtain an exact convex hull for a set of points on two-dimensional space. We call them verified algorithms. We improve a criterion for a fast verified algorithm which is called ’filter’. Finally, numerical examples show that computing times for verified algorithms are comparable to those for non-verified algorithms in many cases.
A Fast Automatic Integration Algorithm using Double Exponential Formula based on Verification Theory
Naoya Yamanaka,Tomoaki Okayama,Shin’ichi Oishi,Takeshi Ogita 대한전자공학회 2009 ITC-CSCC :International Technical Conference on Ci Vol.2009 No.7
A fast automatic integration algorithm of calculating univariate integrals over finite interval using numerical computations is proposed. The proposed algorithm is based on the double exponential formula. In order that the formula works accurately and efficiently, we have presented a theorem for verified computations. In this paper, we propose an approximate integration algorithm based on this theorem. Numerical results are presented showing the performance of the proposed algorithm.