DOCSLIB.ORG

Computer Algebra and Symbolic Computationcohen

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computer Algebra and Symbolic Computationcohen Cohen Computer Algebra and Symbolic Computation and Symbolic Algebra Computer JOEL S. COHEN JOEL S. COHEN Computer Algebra and Symbolic Computation Computer Algebra and Mathematical Methods Symbolic Computation Mathematica™, Maple™, and similar software packages provide Mathematical Methods programs that carry out sophisticated mathematical operations. In this book the author explores the mathematical methods that form the basis for such programs, in particular the application of algorithms to methods such as automatic simplification, polynomial decomposition, and polynomial factorization. Computer Algebra and Symbolic Computation: Mathematical Methods goes beyond the basics of computer algebra—presented in Computer Algebra and Symbolic Computation: Elementary Algorithms—to explore complexity analysis of algorithms and recent developments in the field. This text: • is well-suited for self-study and can be used as the basis for a graduate course. • maintains the style set by Elementary Algorithms and explains mathematical methods as needed. • introduces advanced methods to treat complex operations. • presents implementations in such programs as Mathematica™ and Maple™. Mathematical Methods • includes a CD with the complete text, hyperlinks, and algorithms as well as additional reference files. For the student, Mathematical Methods is an essential companion to Elementary Algorithms, illustrating applications of basic ideas. For the professional, Mathematical Methods is a look at new applications of familiar concepts. ISBN 1-56881-159-4 A K A K Peters, Ltd. ì<(sl&q)=ibbfji< +^-Ä-U-Ä-U PETERS Computer Algebra and Symbolic Computation Computer Algebra and Symbolic Computation Mathematical Methods Joel S. Cohen Department of Computer Science University of Denver A K Peters Natick, Massachusetts Editorial, Sales, and Customer Service Office A K Peters, Ltd. 63 South Avenue Natick, MA 01760 www.akpeters.com Copyright © 2003 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Cohen, Joel S. Computer algebra and symbolic computation : mathematical methods / Joel S. Cohen p. cm. Includes bibliographical references and index. ISBN 1-56881-159-4 1. AlgebraData processing. I. Title. QA155.7.E4 .C635 2002 512dc21 2002024315 Printed in Canada 07 06 05 04 03 10 9 8 7 6 5 4 3 2 1 For my wife Kathryn vii Contents 1Preface ix 1 Background Concepts 1 1.1ComputerAlgebraSystems.................... 1 1.2 Mathematical Pseudo-Language (MPL) . 2 1.3 Automatic Simplification and Expression Structure . 5 1.4 General Polynomial Expressions . 11 1.5MiscellaneousOperators ..................... 12 2 Integers, Rational Numbers, and Fields 17 2.1TheIntegers............................ 17 2.2RationalNumberArithmetic................... 37 2.3Fields................................ 44 3 Automatic Simplification 63 3.1 The Goal of Automatic Simplification . 63 3.2 An Automatic Simplification Algorithm . 91 4 Single Variable Polynomials 111 4.1 Elementary Concepts and Polynomial Division . 111 4.2 Greatest Common Divisors in F[x] . 126 4.3 Computations in Elementary Algebraic Number Fields . 146 4.4PartialFractionExpansioninF(x)................ 166 viii 5 Polynomial Decomposition 179 5.1TheoreticalBackground.................... 180 5.2ADecompositionAlgorithm ................. 188 6 Multivariate Polynomials 201 6.1 Multivariate Polynomials and Integral Domains . 201 6.2 Polynomial Division and Expansion . 207 6.3GreatestCommonDivisors.................. 229 7TheResultant 265 7.1TheResultantConcept.................... 265 7.2 Polynomial Relations for Explicit Algebraic Numbers . 289 8 Polynomial Simplification with Side Relations 297 8.1MultipleDivisionandReduction............... 297 8.2 Equivalence, Simplification, and Ideals . 318 8.3ASimplificationAlgorithm..................334 9 Polynomial Factorization 349 9.1 Square-Free Polynomials and Factorization . 350 9.2 Irreducible Factorization: The Classical Approach . 360 9.3 Factorization in Zp[x]..................... 370 9.4 Irreducible Factorization: A Modern Approach . 399 Bibliography 431 Index 441 Q rshpr ]B8}Z sjAs $R Y hjI BP 8sY8s${R stI {B8}Z R{$t{ Ys $R {Bt{tI b$Y Y I%jB}8t $8}j8ts$Bt stI s}}j${s$Bt BP sjBT $Y8R Ys 8st$}Zjs stI stsjk9 8sY8s${sj -}RR$BtR qY$R ABB, stI Y {B8}st$Bt - ]B8}Z VjAs stI 7k8ABj${ ]B8}Zs$BtU ej8tsk VjB$Y8R s st $tBIZ{$Bt B Y RZA`{ Ys sIIRRR ABY $R }s{${sj stI YB${sj sR}{R ej8tsk VjB$Y8R sIIRRR Y }s{${sj R$I $ $R {Bt{tI b$Y Y PB8Zjs$Bt BP sjB$Y8R Ys RBj% Rk8ABj${ 8sY8s${sj }BAj8R stI b$Y Y $8}j8ts$Bt BP YR sjB$Y8R $t 8R BP Y B}s$BtR stI {BtBj RZ{ZR s%s$jsAj $t {B8}Z sjAs }Bs88$t jstZsR qY$R ABB, bY${Y sIIRRR 8B YB${sj $RRZR $R {Bt{tI b$Y Y AsR${ 8sY8s${sj stI sjBT $Y8${ {Bt{}R Ys s Y PBZtIs$Bt BP Y RZA`{ BY ABB,R R% sR s A$I Abt -R stI 8stZsjR Ys RYBb YBb B ZR {B8}Z sjAs RBPbs stI sIZs j%j -R Ys IR{$A sjB$Y8R s Y PBPBt BP Y hjI qYR ABB,R Ys% At $t %s$BZR RsR BP I%jB}8t PB B% dx ksR qYk s AsRI Bt Y {jsRR tBR PB s bBTJZs {BZR RJZt{ $t {B8}Z sjAs Ys YsR At B&I s Y =t$%R$k BP *t% %k BY ks PB Y }sR dE ksR qY hR {BZR bY${Y $R Y AsR$R PB ejT 8tsk VjB$Y8R ss{R }$8s$jk ZtIsIZs RZItR stI s Pb sIZs RZItR PB8 8sY8s${R {B8}Z R{$t{ stI t$t$t qY R{BtI {BZR bY${Y $R Y AsR$R PB sY8s${sj YBIR ss{R }$8s$jk sIZs RZItR $t ABY 8sY8s${R stI {B8}Z R{$t{ qY {BZR $R {BRRTj$RI ZtI ABY 8sY8s${R stI {B8}Z R{$t{ L[ [ 3UHIDFH Q r rvvr qY s sZI$t{ PB YR ABB,R $t{jZIR RZItR stI }BPRR$BtsjR PB8 8sY8s${R {B8}Z R{$t{ stI BY {Yt${sj hjIR bYB bBZjI j$, B ,tBb sABZ {B8}Z sjAs stI $R s}}j${s$BtR yt Y R}$$ BP st $tBIZ{Bk - b Ys% $I B 8$t$8$9 Y }JZ$R$R qY 8sY8s${sj }JZ$R$R $t{jZI Y ZRZsj bB ks PRY8stuRB}YB8B RJZt{ BP {BZRR w{sj{ZjZR YBZY 8Zj$%s$sAj {sj{ZjZR j8tsk j$ts sjAs stI s}}j$I BI$tsk I$&t$sj JZsT $BtRD yt sII$$Bt st $tBIZ{Bk {BZR $t I$R{ 8sY8s${R $R {T B88tII A{sZR 8sY8s${sj $tIZ{$Bt $R ZRI sR s }BBP {Yt$JZ YBZYBZ qB}${R PB8 j8tsk tZ8A YBk stI sARs{ sjAs s $tBIZ{I sR tII !t Y {B8}Z R{$t{ R$I b sRRZ8 Ys Y sI YsR YsI RB8 -}$t{ b$Y s {B8}Z }Bs88$t jstZs RZ{Y sR 3Bst _sR{sj ] ]LL B Fs%s VjYBZY YR jstZsR s tB ZRI $t YR ABB,R Y R,$jjR $t }BAj8 RBj%$t stI sjB$Y8 I%jB}8t BAs$tI $t s AT $tt$t }Bs88$t {BZR s RRt$sj !t }Bs88$t {Yt$JZ Ys $R R}{$sjjk $8}Bst $t {B8}Z sjAs $R {ZR$Bt VjYBZY 8stk RZItR b$jj Ys% Rt {ZR$Bt $t s {Bt%t$Btsj }Bs88$t {BZR Y B}${ $R IR{$AI $t ]Ys} x BP ej8tsk VjB$Y8R PB8 s {B8}Z sjAs }R}{$% isj$R${sjjk R}s,$t bY$j YR }JZ$R$R RZ^{ $t s PB8sj RtR PB ABY ABB,R $t s }s{${sj RtR Y s RB8 R{$BtR sR Y -R }BRR bY s 8sY8s${sj stI {B8}Zs$Btsj RB}Y$R${s$Bt $R JZ$I VjYBZY Y 8sY8s${sj I%jB}8t $t YR R{$BtR {st A {Ysjjt$t PB RZItR b$Y Y 8$t$8Z8 }JZ$R$R Y sjB$Y8R s s{{RR$Aj stI YR R{$BtR }B%$I s stR$$Bt B 8B sI%st{I s8tR BP Y RZA`{ P thvhv hq 8r BsIjk R}s,$t YR ABB,R s $ttII B R% bB w{B8}j8tskD }Z}BRRU T qB }B%$I s RkR8s${ s}}Bs{Y B Y sjB$Y8${ PB8Zjs$Bt stI $8}j8ts$Bt BP 8sY8s${sj B}s$BtR $t s {B8}Z sjAs }Bs88$t jstZs VjB$Y8${ 8YBIR $t sI$$Btsj 8sY8s${R s ZRZsjjk tB }T RtI b$Y Y }{$R$Bt PBZtI $t tZ8${sj 8sY8s${R B {Bt%t$Btsj {B8}Z }Bs88$t 3B -s8}j Y sjB$Y8 PB Y -}stR$Bt BP }BIZ{R stI }BbR BP }BjktB8$sjR $R ZRZsjjk $%t $tPB8sjjk $tRsI BP b$Y w{ZR$%D }B{IZR Ys {st A -}RRI sR s {B8}Z }Bs8 3UHIDFH [L qY 8s$sj $t ej8tsk VjB$Y8R $R {Bt{tI b$Y Y sjB$YT 8${ PB8Zjs$Bt BP RBjZ$BtR B j8tsk Rk8ABj${ 8sY8s${sj }BAT j8R qY %$b}B$t $R Ys 8sY8s${sj -}RR$BtR }RtI sR -T }RR$Bt R s Y Iss BA`{R BP {B8}Z sjAs }Bs8R stI ZR$t s Pb }$8$$% B}s$BtR Ys stsjk9 stI {BtRZ{ -}RR$BtR b {st $8}j8t 8stk j8tsk B}s$BtR PB8 sjAs $BtB8k {sj{ZjZR stI I$&t$sj JZs$BtR 3B -s8}j sjB$Y8R s $%t PB Y stsjkR$R stI 8st$}Zjs$Bt BP }BjktB8$sjR stI s$Btsj -}RR$BtR Y 8st$}Zjs$Bt BP -}Btt$sj stI $BtB8${ PZt{$BtR I$&t$s$Bt j8tsk $ts$Bt stI Y RBjZ$Bt BP hR BI I$&t$sj JZsT $BtR BR BP Y 8s$sj $t Y$R ABB, $R tB PBZtI $t $Y 8sY8s${R -ABB,R B $t BY 8B sI%st{I {B8}Z sjAs -ABB,R T qB IR{$A RB8 BP Y 8sY8s${sj {Bt{}R stI sjB$Y8${ {YT t$JZR Z$j$9I Ak 8BIt {B8}Z sjAs RBPbs 3B Y }sR nx ksR Y Rs{Y $t {B8}Z sjAs YsR At {BtT {tI b$Y Y I%jB}8t BP &{$% stI ^{$t sjB$Y8R PB 8stk 8sY8s${sj B}s$BtR $t{jZI$t }BjktB8$sj sR {B88Bt I$%$RB w{ID {B8}Zs$Bt }BjktB8$sj Ps{B$9s$Bt }BjktB8$sj I{B8}BR$$Bt Y RBjZ$Bt BP RkR8R BP j$ts JZs$BtR stI 8Zj$%s$s }BjktB8$sj JZs$BtR $tIht$ $ts$Bt stI Y RBjZ$Bt BP I$&t$sj JZs$BtR VjYBZY sjB$Y8R PB RB8 BP YR }BAj8R Ys% At ,tBbt R$t{ Y t$ttY {tZk PB ^{$t{k sRBtR Yk s tB RZ$sAj sR tsj }Z}BR sjB$Y8R PB {B8}Z sjAs RBPbs qY {jsRR${sj sjB$Y8R s $8}Bst YBb% A{sZR Yk s 8Z{Y R$8}j stI }B%$I s {BtT - B 8B$%s Y AsR${ sjAs${ $IsR stI Y tI PB 8B ^{$t s}}Bs{YR qY 8s$sj $t sY8s${sj YBIR $R st $tBIZ{$Bt B Y 8sYT 8s${sj {Yt$JZR stI sjB$Y8${ 8YBIR BP {B8}Z sjAs VjT YBZY Y 8s$sj $t Y$R ABB, $R 8B I$^{Zj stI JZ$R s 8sYT 8s${sj RB}Y$R${s$Bt Y s}}Bs{Y stI Rj{$Bt BP B}${R $R IR$tI RB Ys $ $R s{{RR$Aj stI $tR$t B Y $ttII sZI$t{ VjB$Y8R s $%t PB AsR${ $t stI s$Btsj tZ8A B}s$BtR sZB8s${ wB IPsZjD R$8}j$h{s$Bt BP sjAs${ -}RR$BtR sR {B88Bt I$%$RB {sj{Zjs$Bt PB R$tj stI 8Zj$%s$s }BjktB8$sjR RZjst {B8}Zs$Bt }BjktB8$sj I{B8}BR$$Bt }BjktB8$sj R$8}j$h{s$Bt b$Y p[BAt AsRR stI }BjktB8$sj Ps{B$9s$Bt Uvp Tryrpv qY sZYB BP st $tBIZ{Bk - sABZ s s}$Ijk {Yst$t hjI $R Ps{I b$Y s I$^{Zj I{$R$Bt sABZ bY${Y B}${R stI sjB$Y8R B $t{jZI $t s [LL 3UHIDFH - qY$R I{$R$Bt $R {BtRs$tI Ak Y As{,BZtI BP Y sZI$t{ Y 8sY8s${sj I$^{Zjk BP Y 8s$sj stI BP {BZR Ak R}s{ j$8$s$BtR yt sII$$Bt b Aj$% Ys st $tBIZ{Bk - RYBZjI sjjk A st $tBT
Load more

Recommended publications
  • Some Reflections About the Success and Impact of the Computer Algebra System DERIVE with a 10-Year Time Perspective
    Noname manuscript No. (will be inserted by the editor) Some reflections about the success and impact of the computer algebra system DERIVE with a 10-year time perspective Eugenio Roanes-Lozano · Jose Luis Gal´an-Garc´ıa · Carmen Solano-Mac´ıas Received: date / Accepted: date Abstract The computer algebra system DERIVE had a very important im- pact in teaching mathematics with technology, mainly in the 1990's. The au- thors analyze the possible reasons for its success and impact and give personal conclusions based on the facts collected. More than 10 years after it was discon- tinued it is still used for teaching and several scientific papers (most devoted to educational issues) still refer to it. A summary of the history, journals and conferences together with a brief bibliographic analysis are included. Keywords Computer algebra systems · DERIVE · educational software · symbolic computation · technology in mathematics education Eugenio Roanes-Lozano Instituto de Matem´aticaInterdisciplinar & Departamento de Did´actica de las Ciencias Experimentales, Sociales y Matem´aticas, Facultad de Educaci´on,Universidad Complutense de Madrid, c/ Rector Royo Villanova s/n, 28040-Madrid, Spain Tel.: +34-91-3946248 Fax: +34-91-3946133 E-mail: [email protected] Jose Luis Gal´an-Garc´ıa Departamento de Matem´atica Aplicada, Universidad de M´alaga, M´alaga, c/ Dr. Ortiz Ramos s/n. Campus de Teatinos, E{29071 M´alaga,Spain Tel.: +34-952-132873 Fax: +34-952-132766 E-mail: [email protected] Carmen Solano-Mac´ıas Departamento de Informaci´ony Comunicaci´on,Universidad de Extremadura, Plaza de Ibn Marwan s/n, 06001-Badajoz, Spain Tel.: +34-924-286400 Fax: +34-924-286401 E-mail: [email protected] 2 Eugenio Roanes-Lozano et al.
    [Show full text]
  • Redberry: a Computer Algebra System Designed for Tensor Manipulation
    Redberry: a computer algebra system designed for tensor manipulation Stanislav Poslavsky Institute for High Energy Physics, Protvino, Russia SRRC RF ITEP of NRC Kurchatov Institute, Moscow, Russia E-mail: [email protected] Dmitry Bolotin Institute of Bioorganic Chemistry of RAS, Moscow, Russia E-mail: [email protected] Abstract. In this paper we focus on the main aspects of computer-aided calculations with tensors and present a new computer algebra system Redberry which was specifically designed for algebraic tensor manipulation. We touch upon distinctive features of tensor software in comparison with pure scalar systems, discuss the main approaches used to handle tensorial expressions and present the comparison of Redberry performance with other relevant tools. 1. Introduction General-purpose computer algebra systems (CASs) have become an essential part of many scientific calculations. Focusing on the area of theoretical physics and particularly high energy physics, one can note that there is a wide area of problems that deal with tensors (or more generally | objects with indices). This work is devoted to the algebraic manipulations with abstract indexed expressions which forms a substantial part of computer aided calculations with tensors in this field of science. Today, there are many packages both on top of general-purpose systems (Maple Physics [1], xAct [2], Tensorial etc.) and standalone tools (Cadabra [3,4], SymPy [5], Reduce [6] etc.) that cover different topics in symbolic tensor calculus. However, it cannot be said that current demand on such a software is fully satisfied [7]. The main difference of tensorial expressions (in comparison with ordinary indexless) lies in the presence of contractions between indices.
    [Show full text]
  • Software for Numerical Computation
    Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1977 Software for Numerical Computation John R. Rice Purdue University, [email protected] Report Number: 77-214 Rice, John R., "Software for Numerical Computation" (1977). Department of Computer Science Technical Reports. Paper 154. https://docs.lib.purdue.edu/cstech/154 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. SOFTWARE FOR NUMERICAL COMPUTATION John R. Rice Department of Computer Sciences Purdue University West Lafayette, IN 47907 CSD TR #214 January 1977 SOFTWARE FOR NUMERICAL COMPUTATION John R. Rice Mathematical Sciences Purdue University CSD-TR 214 January 12, 1977 Article to appear in the book: Research Directions in Software Technology. SOFTWARE FOR NUMERICAL COMPUTATION John R. Rice Mathematical Sciences Purdue University INTRODUCTION AND MOTIVATING PROBLEMS. The purpose of this article is to examine the research developments in software for numerical computation. Research and development of numerical methods is not intended to be discussed for two reasons. First, a reasonable survey of the research in numerical methods would require a book. The COSERS report [Rice et al, 1977] on Numerical Computation does such a survey in about 100 printed pages and even so the discussion of many important fields (never mind topics) is limited to a few paragraphs. Second, the present book is focused on software and thus it is natural to attempt to separate software research from numerical computation research. This, of course, is not easy as the two are intimately intertwined.
    [Show full text]
  • CAS (Computer Algebra System) Mathematica
    CAS (Computer Algebra System) Mathematica- UML students can download a copy for free as part of the UML site license; see the course website for details From: Wikipedia 2/9/2014 A computer algebra system (CAS) is a software program that allows [one] to compute with mathematical expressions in a way which is similar to the traditional handwritten computations of the mathematicians and other scientists. The main ones are Axiom, Magma, Maple, Mathematica and Sage (the latter includes several computer algebras systems, such as Macsyma and SymPy). Computer algebra systems began to appear in the 1960s, and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martin Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for "clean ship") in 1963. Using LISP as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 Systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH-Emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory") which is a system for numerical computation built 15 years later at the University of New Mexico, accidentally named rather similarly. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained.
    [Show full text]
  • Quantum Hypercomputation—Hype Or Computation?
    Quantum Hypercomputation—Hype or Computation? Amit Hagar,∗ Alex Korolev† February 19, 2007 Abstract A recent attempt to compute a (recursion–theoretic) non–computable func- tion using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from. ∗HPS Department, Indiana University, Bloomington, IN, 47405. [email protected] †Philosophy Department, University of BC, Vancouver, BC. [email protected] 1 1 Introduction Combining physics, mathematics and computer science, quantum computing has developed in the past two decades from a visionary idea (Feynman 1982) to one of the most exciting areas of quantum mechanics (Nielsen and Chuang 2000). The recent excitement in this lively and fashionable domain of research was triggered by Peter Shor (1994) who showed how a quantum computer could exponentially speed–up classical computation and factor numbers much more rapidly (at least in terms of the number of computational steps involved) than any known classical algorithm. Shor’s algorithm was soon followed by several other algorithms that aimed to solve combinatorial and algebraic problems, and in the last few years the- oretical study of quantum systems serving as computational devices has achieved tremendous progress. According to one authority in the field (Aharonov 1998, Abstract), we now have strong theoretical evidence that quantum computers, if built, might be used as powerful computational tool, capable of per- forming tasks which seem intractable for classical computers.
    [Show full text]
  • Analysis and Verification of Arithmetic Circuits Using Computer Algebra Approach
    University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses March 2020 ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH TIANKAI SU University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the VLSI and Circuits, Embedded and Hardware Systems Commons Recommended Citation SU, TIANKAI, "ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH" (2020). Doctoral Dissertations. 1868. https://doi.org/10.7275/15875490 https://scholarworks.umass.edu/dissertations_2/1868 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH A Dissertation Presented by TIANKAI SU Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY February 2020 Electrical and Computer Engineering c Copyright by Tiankai Su 2020 All Rights Reserved ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH A Dissertation Presented by TIANKAI SU Approved as to style and content by: Maciej Ciesielski, Chair George S. Avrunin, Member Daniel Holcomb, Member Weibo Gong, Member Christopher V. Hollot, Department Head Electrical and Computer Engineering ABSTRACT ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH FEBRUARY 2020 TIANKAI SU B.Sc., NORTHEAST DIANLI UNIVERSITY Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Maciej Ciesielski Despite a considerable progress in verification of random and control logic, ad- vances in formal verification of arithmetic designs have been lagging.
    [Show full text]
  • Mathematics and Computation
    Mathematics and Computation Mathematics and Computation Ideas Revolutionizing Technology and Science Avi Wigderson Princeton University Press Princeton and Oxford Copyright c 2019 by Avi Wigderson Requests for permission to reproduce material from this work should be sent to [email protected] Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved Library of Congress Control Number: 2018965993 ISBN: 978-0-691-18913-0 British Library Cataloging-in-Publication Data is available Editorial: Vickie Kearn, Lauren Bucca, and Susannah Shoemaker Production Editorial: Nathan Carr Jacket/Cover Credit: THIS INFORMATION NEEDS TO BE ADDED WHEN IT IS AVAILABLE. WE DO NOT HAVE THIS INFORMATION NOW. Production: Jacquie Poirier Publicity: Alyssa Sanford and Kathryn Stevens Copyeditor: Cyd Westmoreland This book has been composed in LATEX The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper 1 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Dedicated to the memory of my father, Pinchas Wigderson (1921{1988), who loved people, loved puzzles, and inspired me. Ashgabat, Turkmenistan, 1943 Contents Acknowledgments 1 1 Introduction 3 1.1 On the interactions of math and computation..........................3 1.2 Computational complexity theory.................................6 1.3 The nature, purpose, and style of this book............................7 1.4 Who is this book for?........................................7 1.5 Organization of the book......................................8 1.6 Notation and conventions.....................................
    [Show full text]
  • Computer Algebra Tems
    short and expensive. Consequently, computer centre managers were not easily persuaded to install such sys­ Computer Algebra tems. This is another reason why many specialized CAS have been developed in many different institutions. From the Jacques Calmet, Grenoble * very beginning it was felt that the (UFIA / INPG) breakthrough of Computer Algebra is linked to the availability of adequate computers: several megabytes of main As soon as computers became avai­ handling of very large pieces of algebra memory and very large capacity disks. lable the wish to manipulate symbols which would be hopeless without com­ Since the advent of the VAX's and the rather than numbers was considered. puter aid, the possibility to get insight personal workstations, this era is open­ This resulted in what is called symbolic into a calculation, to experiment ideas ed and indeed the use of CAS is computing. The sub-field of symbolic interactively and to save time for less spreading very quickly. computing which deals with the sym­ technical problems. When compared to What are the main CAS of possible bolic solution of mathematical problems numerical computing, it is also much interest to physicists? A fairly balanced is known today as Computer Algebra. closer to human reasoning. answer is to quote MACSYMA, RE­ The discipline was rapidly organized, in DUCE, SMP, MAPLE among the general 1962, by a special interest group on Computer Algebra Systems purpose ones and SCHOONSHIP, symbolic and algebraic manipulation, Computer Algebra systems may be SHEEP and CAYLEY among the specia­ SIGSAM, of the Association for Com­ classified into two different categories: lized ones.
    [Show full text]
  • Mathematical Modeling with Technology PURNIMA RAI
    International Journal of Innovative Research in Computer Science & Technology (IJIRCST) ISSN: 2347-5552, Volume-3, Issue-2, March 2015 Mathematical Modeling with Technology PURNIMA RAI Abstract— Mathematics is very important for an Computer Algebraic System (CAS) increasing number of technical jobs in a quantitative A Computer Algebra system is a type of software package world. Over recent years technology has progressed to that is used in manipulation of mathematical formulae. The the point where many computer algebraic systems primary goal of a Computer Algebra system is to automate (CAS) are widely available. These technologies can tedious and sometimes difficult algebraic manipulation perform a wide variety of algebraic tasks. Computer Algebra systems often include facilities for manipulations.Computer algebra systems (CAS) have graphing equations and provide a programming language become powerful and flexible tools to support a variety for the user to define their own procedures. Computer of mathematical activities. This includes learning and Algebra systems have not only changed how mathematics is teaching, research and assessment. MATLAB, an taught at many universities, but have provided a flexible acronym for “Matrix Laboratory”, is the most tool for mathematicians worldwide. Examples of popular extensively used mathematical software in the general systems include Maple, Mathematica, and MathCAD. sciences. MATLAB is an interactive system for Computer Algebra systems can be used to simplify rational numerical computation and a programmable system. functions, factor polynomials, find the solutions to a system This paper attempts to investigate the impact of of equation, and various other manipulations. In Calculus, integrating mathematical modeling and applications on they can be used to find the limit of, symbolically integrate, higher level mathematics in India.
    [Show full text]
  • The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems
    The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them? Antonio J. Durán, Mario Pérez, and Juan L. Varona Introduction by a typical research mathematician, but let us Nowadays, mathematicians often use a computer explain it briefly. It is not necessary to completely algebra system as an aid in their mathematical understand the mathematics, just to realize that it research; they do the thinking and leave the tedious is typical mathematical research using computer calculations to the computer. Everybody “knows” algebra as a tool. that computers perform this work better than Our starting point is a discrete positive measure people. But, of course, we must trust in the results on the real line, µ n 0 Mnδan (where δa denotes = ≥ derived via these powerful computer algebra the Dirac delta at a, and an < an 1) having P + systems. First of all, let us clarify that this paper is a sequence of orthogonal polynomials Pn n 0 f g ≥ not, in any way, a comparison between different (where Pn has degree n and positive leading computer algebra systems, but a sample of the coefficient). Karlin and Szeg˝oconsidered in 1961 (see [4]) the l l Casorati determinants current state of the art of what mathematicians × can expect when they use this kind of software. Pn(ak)Pn(ak 1):::Pn(ak l 1) Although our example deals with a concrete system, + + − Pn 1(ak)Pn 1(ak 1):::Pn 1(ak l 1) 0 + + + + + − 1 we are sure that similar situations may occur with (1) det .
    [Show full text]
  • Addition and Subtraction
    A Addition and Subtraction SUMMARY (475– 221 BCE), when arithmetic operations were per- formed by manipulating rods on a flat surface that was Addition and subtraction can be thought of as a pro- partitioned by vertical and horizontal lines. The num- cess of accumulation. For example, if a flock of 3 sheep bers were represented by a positional base- 10 system. is joined with a flock of 4 sheep, the combined flock Some scholars believe that this system— after moving will have 7 sheep. Thus, 7 is the sum that results from westward through India and the Islamic Empire— the addition of the numbers 3 and 4. This can be writ- became the modern system of representing numbers. ten as 3 + 4 = 7 where the sign “+” is read “plus” and The Greeks in the fifth century BCE, in addition the sign “=” is read “equals.” Both 3 and 4 are called to using a complex ciphered system for representing addends. Addition is commutative; that is, the order numbers, used a system that is very similar to Roman of the addends is irrelevant to how the sum is formed. numerals. It is possible that the Greeks performed Subtraction finds the remainder after a quantity is arithmetic operations by manipulating small stones diminished by a certain amount. If from a flock con- on a large, flat surface partitioned by lines. A simi- taining 5 sheep, 3 sheep are removed, then 2 sheep lar stone tablet was found on the island of Salamis remain. In this example, 5 is the minuend, 3 is the in the 1800s and is believed to date from the fourth subtrahend, and 2 is the remainder or difference.
    [Show full text]
  • Arxiv:1503.05401V1 [Math.NT] 18 Mar 2015 W Teswt H Aecmoiin Itte Ovdteequation the Solved Then Ritt Composition
    FUNCTIONAL COMPOSITION OF POLYNOMIALS: INDECOMPOSABILITY, DIOPHANTINE EQUATIONS AND LACUNARY POLYNOMIALS DIJANA KRESO AND ROBERT F. TICHY Dedicated to Professor Ludwig Reich on the occasion of his 75th birthday Abstract. Starting from Ritt’s classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x)= g(y). 1. Introduction In 1920’s, in the frame of investigations of functional equations by the founders of modern iteration theory (Fatou, Julia and Ritt), Ritt [53] studied equations of type f1 ◦ f2 ◦···◦ fm = g1 ◦ g2 ◦···◦ gn in nonconstant complex polynomials. This resulted in him studying the possible ways of writing a complex polynomial as a functional composition of polynomials of lower degree. For an arbitrary field K, a polynomial f ∈ K[x] with deg f > 1 is called inde- composable (over K) if it cannot be written as the composition f(x)= g(h(x)) with g,h ∈ K[x] and deg g > 1, deg h > 1. Any representation of f(x) as a functional composition of polynomials of degree greater than 1 is said to be a decomposition arXiv:1503.05401v1 [math.NT] 18 Mar 2015 of f(x). It follows by induction that any polynomial f(x) with deg f > 1 can be written as a composition of indecomposable polynomials – such an expression for f(x) is said to be a complete decomposition of f(x). A complete decomposition of a polynomial clearly always exists, but it does not need to be unique.
    [Show full text]