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A review note on arbitrary precision arithmetic

Szczygieł Bartłomiej, Marcinkowski Leszek

Mathematica Applicanda

2022

Vol. 50, no. 2

153--164

In this paper, we present a note on arbitrary precision. We give two simple examples showing the need of using arbitrary precision arithmetic. Next, we discuss how to use arbitrary precision arithmetic types in MATLAB/OCTAVE and further present short descriptions of several basic, in particular C/C++, packages for using arbitrary precision arithmetic in numerical codes for scientific computations. Finally, we discuss the contribution of one of the authors in the development of a library for arbitrary precision floating point numbers briefly.

W tej pracy przedstawiamy krótką przeglądową notkę na temat arytmetyki zmiennopozycyjnej dowolnej precyzji. Przedstawiamy dwa proste przykłady pokazuj¡ce konieczność wykorzystania takiej arytmetyki. Omawiamy użycie arytmetyki zmiennopozycyjnej dowolnej precyzji w MATLABie-/Octave'ie i dalej omawiamy krótko kilka podstawowych, w szczególności w C/C++, bibliotek zawierających arytmetyke dowolnej precyzji dla kodów (programów) numerycznych dla zastosowań w obliczeniach naukowych. Ostatecznie omawiamy wkład jednego z autorów w rozwój biblioteki dla arytmetyki dowolnej precyzji.

An Introduction to the Special Issue on Numerical Techniques Meet with OR - Part II

Gürbüz Burcu, Weber Gerhard-Wilhelm

Foundations of Computing and Decision Sciences

2021

Vol. 46, No. 3

201--204

The special issue: “Numerical Techniques Meet with OR” of the Foundations of Computing and Decision Sciences consists of two parts which are of the main theme of numerical techniques and their applications in multi-disciplinary areas. The first part of this special issue was already collected in the FCDS Vol. 46, issue 1. In this second part of our special issue editorial, a description of the special issue presents numerical methods which can be used as alternative techniques for Scientific Computing and led Operational Research applications in many fields for further investigation.

Efficient Computation of the Large Inductive Dimension Using Order- and Graph-theoretic Means

Berghammer Rudolf, Schnoor Henning, Winter Michael

Fundamenta Informaticae

2020

Vol. 177, nr 2

95--113

Finite topological spaces and their dimensions have many applications in computer science, e.g., in digital topology, computer graphics and the analysis and synthesis of digital images. Georgiou et. al. [11] provided a polynomial algorithm for computing the covering dimension dim (X, 𝒯 ) of a finite topological space (X, 𝒯 ). In addition, they asked whether algorithms of the same complexity for computing the small inductive dimension ind (X, 𝒯 ) and the large inductive dimension Ind (X, 𝒯 ) can be developed. The first problem was solved in a previous paper [4]. Using results of the that paper, we also solve the second problem in this paper. We present a polynomial algorithm for Ind (X, 𝒯 ), so that there are now efficient algorithms for the three most important notions of a dimension in topology. Our solution reduces the computation of Ind (X, 𝒯 ), where the specialisation pre-order of (X, 𝒯 ) is taken as input, to the computation of the maximal height of a specific class of directed binary trees within the partially ordered set. For the latter an efficient algorithm is presented that is based on order- and graph-theoretic ideas. Also refinements and variants of the algorithm are discussed.