In the following, we describe in detail the current implementation of Taylor model arithmetic in version 8.1 of the code COSY INFINITY. Since in the Taylor model approach, the coefficients are floating point (FP) numbers, care must be taken that the inaccuracies of conventional FP arithmetic are properly accounted for. Algorithmically the methods are rather straightforward; however for practica...

We explore what it means to apply automatic diierentiation with respect to a set of real variables to codes containing complex arithmetic. That is, both dependent and independent variables with respect to diierentiation are real variables, but in order to exploit features of complex mathematics, part of the code is expressed by employing complex arithmetic. We investigate how one can apply auto...

A newly developed grammar-based lossless source coding theory and its implementation was proposed in 1999 and 2000, respectively, by Yang and Kieffer. The code first transforms the original data sequence into an irreducible context-free grammar, which is then compressed using arithmetic coding. In the study of grammarbased coding for mammography applications, we encountered two issues: processi...

We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-di...

The FRPOLY Lisp performance benchmark [3] was based on a circa1968 piece of code for computing powers of polynomials. We address two questions: (a) What algorithm would you use if you really wanted to compute powers of polynomials fast? and (b) Given that Common Lisp supports many types of data structures other than the simple lists used for the benchmark, what more efficient representations mi...

Floating-point arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32-bit (respectively 64-bit) long. Extended precision formats can be up to 128-bit long. However some problems require a longer floating-point format, because of round-off errors. Such problems are usually solved in arbitrary precision, but round-off errors still occur and must be ...

Cryptography has been used from time immemorial for preserving the confidentiality of data/information in storage or transit. Thus, cryptography research also evolving classical Caesar cipher to modern cryptosystems, based on modular arithmetic contemporary cryptosystems quantum computing. The emergence computing poses a major threat arithmetic, whereby even computationally hard problems which ...

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...

Motivated by recent results in Joint Source/Channel (JSC) coding and decoding, this paper addresses the problem of soft input decoding of Arithmetic Codes (AC). A new length-constrained scheme for JSC decoding of these codes is proposed based on the Maximum a posteriori (MAP) sequence estimation criterion. The new decoder, called Chase-like arithmetic decoder is supposed to know the source symb...

This thesis presents a robust method for evaluating differential geometry properties of sculptured surfaces by using a validated ordinary differential equation (ODE) system solver based on interval arithmetic. Iso-contouring of curvature of a Bezier surface patch, computation of curvature lines of a Bezier surface patch and computation of geodesics of a Bezier surface patch are computed by the ...