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Lecture Notes in Computer Science
Informatique théorique et applications, tome 23, n o 1 (1989), p. 101-111. <http © AFCET, 1989, tous droits réservés. L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques Informatique théorique et Applications/Theoretical Informaties and Applications (vol.23, n° 1, 1989, p. 101 à 111) ON COMPUTATIOIMS WITH INTEGER DIVISION by Bettina JUST (*), Friedhelm MEYER AUF DER HEIDE ( 2 )
Springer eBooks, 1993
Computation trees with operation set S f+;?; ;DIV; DIV c g (S-CTs) are considered. DIV denotes integer division, DIV c integer division by constants. We characterize the families of languages L Z Z n that can be recognized by S-CTs, separate the computational capabilities of S-CTs for di erent operation sets S, and prove lower bounds for the depth of such trees. Let CC n (S) denote the family of languages L Z Z n that can be recognized by an S-CT. In 7], CC 1 fSg is characterized for all S f+;?; ;DIV; DIV c g. It turns out that CC 1 (f+; ?; DIV c g) = CC 1 f+;?; ;DIVg. In this paper we shed some more light on the computational power of integer division: { We characterize CC n (S), n > 1, for S = f+; ?;DIV c g and S = f+; ?; ; DIV c g, and partially characterize CC n (S); n 1, for S = f+;?; DIVg and S = f+;?; ; DIVg. { We completely determine the relations among the classes CC n (S). We further prove lower bounds: { The component counting lower bound (e. g. (n 2) for the knapsack problem) proven for S = f+;?; g by Ben Or and Yao also holds for f+;?; ; DIV c g. { The GCD-algorithm due to Brent and Kung for f+;?;DIV c g-CTs is optimal even for f+; ?;DIVg-CTs. { Testing whether q(y) > x for an irreducible polynomial q of degree d takes time (loglog(d)) for f+;?; ; DIVg-CTs, even if arbitrary rational constants can be used at unit cost. This is the rst nontrivial lower bound in this strong model (in which e. g. every nite language can be recognized in constant time, independent of the size of the language).
IEEE Transactions on Computers
C OMPUTER arithmetic is used in many applications, usually totally silently (one should keep in mind that even when running programs that are not at all numeric, memory addresses are computed, which involves additions, multiplications, and sometimes divisions). However, in some areas, it plays a central role. To give a few examples:
IEEE Transactions on Computers, 2000
Journal de Théorie des Nombres de Bordeaux, 1998
L'accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
ArXiv, 2010
This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware - we do not cover computer architecture or the design of computer hardware. Instead we focus on algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. The algorithms that we present are mainly intended for arbitrary-precision arithmetic. They are not limited by the computer word size, only by the memory and time available for the computation. We consider both integer and real (floating-point) computations. The book is divided into four main chapters, plus an appendix. Our aim is to present the latest developments in a concise manner. At the same time, we provide a self-contained introduction for...
Proceedings of International Conference on Application Specific Systems, Architectures and Processors: ASAP '96, 1996
We propose in this paper a new algorithm and architecture for performing divisions in residue number systems. Our algorithm is suitable for residue number systems with large moduli, with the aim of manipulating very large integers on a parallel computer or a specialpurpose architecture. The two basic features of our algorithm are one one hand the use of a high-radix division method, and on the other hand the use of a floating-point arithmetic that should run in parallel with the modular arithmetic.
The Mathematics Enthusiast
This paper discusses the formalization of the binary number system and the groundwork that was laid for the future of digital circuitry, computers, and the field of computer science. The goal of this paper is to show how Gottfried Leibniz formalized the binary number system and solidified his thoughts through an analysis of the Chinese I Ching. In addition, Leibniz's work in logic and with computing machines is presented. This work laid the foundation for Boolean algebra and digital circuitry which was continued by George Boole, Augustus De Morgan, and Claude Shannon in the centuries following. Some have coined Leibniz the world's first computer scientist, and this paper will attempt to demonstrate a validation of this conjecture.
1972 IEEE 2nd Symposium on Computer Arithmetic (ARITH), 1972
J.UCS The Journal of Universal Computer Science, 1996
This surveys algorithms and circuits for integer division in special cases. These include division by constants, small divisors, exact divisors, and cases where the divisor and the number base have a special relationship. The related operation of remainder is also covered. Various prior techniques are treated in a common framework. Worked examples are provided together with examples of practical application. Category: B2.0 Arithmetic and Logic Circuits We will assume that integers are presented, using a positive integer base b, as n-digit vectors. In particular we have D, d, q, r, where, for example:
Based on a three week study conducted with students (n ~ 42) at a government girls school in rural Madhya Pradesh, this paper analyses some of the difficulties surrounding the teaching and learning of the standard long division algorithm, and argues that an alternate approach based on sharing makes for a more flexible and effective teaching of division. The work done is preliminary, but offers insights into developing an alternate teaching-learning trajectory for division beginning at the primary school level.
2009
My first experiences with decimal computer arithmetic in college (1963) influenced my subsequent career decisions and projects as described herein. Many popular early computers focused on commercial applications for which decimal arithmetic was appropriate. These digitserial implementations did not minimize hardware cost, and provided the precision needed by the application. Decimal arithmetic as taught in elementary school is a fine starting point for describing computer operation, but for nonengineers the hardware realization of an adder is mysterious. Routing circuits, while not always practical, illustrate how two-bits can be added by using switches, relays, or MOSFET's.
International Journal of Software & Hardware Research in Engineering, 8(2), 27–35., 2020
The article contains a prospectus of the book under the same title [1]. This book is published only in Russian and in this connection, this prospectus is published. The book contains 673 pages. The author seeks assistance in publishing a book in English.
2019
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Introduction to the papers of TWG02: Arithmetic and number systems Sebastian Rezat, Lisser Ejersbo, Darina Jirotkova, Elisabeth Rathgeb-Schnierer
Formal Methods in System Design, 1994
This article describes a proof of the functional correctness of a nonrestoring division algorithm and its implementation on an ALU. The first part of the proof deals with the correctness of the division algorithm with respect to a specification of division on the integer level. The second part is concerned with the correctness of the actual implementation, which is proven by checking several refinements of the algorithm. All the proofs have been mechanically checked with the Boyer-Moore theorem-proving system, in some cases making use of the interactive proof checker for the system.
2010 IEEE International Workshop on Information Forensics and Security, 2010
When processing signals in the encrypted domain, homomorphic encryption can be used to enable linear operations on encrypted data. Integer division of encrypted data however requires an additional protocol with the server and will be relatively expensive. We present new solutions for dividing encrypted data, having low computational complexity. Two protocols for computing exact division, and two for approximating the division result.
IEEE Transactions on Computers, 1977
either by a ring counter or a shift register with a one rippling through. VI. CONCLUSION A BIN/BCD conversion method has been developed which lends itself to unlimited expansion by using the geometrical similarity of interconnecting maps. Properly designed, the maps then contain all the necessary information to determine the size, the content, and the actual wiring of the decoding ROM's. Wiring diagrams were used to develop the hybrid conversion scheme. Conversion systems for practically any speed or size can be de- signed by using either the static or the hybrid method. This scheme is not limited to binary/BCD conversion, but can be ex- tended to other types of conversion as well, such as, synchro/ BCD, etc. On the Use of Continued Fractions for Digital Computer Arithmetic KISHOR S. TRIVEDI Abstract-Recently, there has been some interest in the use of continued fractions for digital hardware calculations. We require that the coefficients of the continued fractions be integral powers of 2 and, therefore, well-known continued fraction expansions of functions cannot be used. Methods of expansion of a large number of functions are presented. We show that the problem of selection of coefficients of the continued fractions does not have practical solution in most of the cases we have considered.
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