Computer Algebra Handbook: Foundations - Applications - Systems

ISBN-13: 9783642629884 / Angielski / Miękka / 2012 / 637 str.

Johannes Grabmeier; Erich Kaltofen; Volker Weispfenning

Opis

Two ideas lie gleaming on the jeweler's velvet. The first is the calculus, the sec ond, the algorithm. The calculus and the rich body of mathematical analysis to which it gave rise made modern science possible; but it has been the algorithm that has made possible the modern world. -David Berlinski, The Advent of the Algorithm First there was the concept of integers, then there were symbols for integers: I, II, III, 1111, fttt (what might be called a sticks and stones representation); I, II, III, IV, V (Roman numerals); 1, 2, 3, 4, 5 (Arabic numerals), etc. Then there were other concepts with symbols for them and algorithms (sometimes) for ma nipulating the new symbols. Then came collections of mathematical knowledge (tables of mathematical computations, theorems of general results). Soon after algorithms came devices that provided assistancefor carryingout computations. Then mathematical knowledge was organized and structured into several related concepts (and symbols): logic, algebra, analysis, topology, algebraic geometry, number theory, combinatorics, etc. This organization and abstraction lead to new algorithms and new fields like universal algebra. But always our symbol systems reflected and influenced our thinking, our concepts, and our algorithms."

Recenzje

From the reviews:

"This is a substantially enlarged report originally covering the state and future of computer algebra in Germany. ... Overall, it gives a high-level (in the sense of abstraction; no theorems, let alone proofs, are in general provided) autoritative overview of what is going on and should be of help to poeple looking for a research topic, software for specific problems or, later on, writing on the history of the subject."

H.Muthsam, Monatshefte für Mathematik 142, issue 2, p. 170, 2004

"This handbook is very impressive. Written by more than two hundred spcialists, this international eidition of a previous German report presents the state of the art of a large field of computational mathematics called computer algebra. .... So, as Bob Caviness says in the preface of the book, "Gentle reader, I recommend this volume and all its concepts, symbols, and algorihtms to you."

J.M.Ollagnier, Mathematical Reviews Clippings from Issue 2004h

"This encyclopaedic book is a wonderful source of information on computer algebra. ... Although most people who come to this book will probably use it as an encyclopaedia, it is much more than that. Researchers in coputer algebra will benefit from the extensive bibliography, and lecturers will find it a mine of ideas for interesting topics that might be discussed in a course on computer algebra and its applications. Finally, to all computer algebra aficionados this book will also provide wonderful entertainment for many a rainy day."

S.C.Coutinho, The Mathematical Gazette Vol. 88, Issue 512 (2004) 410-411

"... A handbook must be considered as well-composed if it addresses "all" issues of the area and if its individual articles provide rough surveys of their subject and point the reader to the relevant recent literature. From this point of view, this Computer Algebra Handbook is certainly well-composed: To strangers of the area it offers easy gathering of information about a particular subject (to be located either through the Table of Contents or the Subject Index"; to experts in computer algebra, it provides access to subjects outside their expertise. ... The team of editors must be congratulated upon succeeding in collecting contributions of more than 200 authors which make a concise and homogeneous volume."

H.Stetter, IMN - Internationale Mathematische Nachrichten 194, 2003

"... The interesting focus of the book is the fact that the editors have been able to encourage about 200 specialsit on different branches of the subject to contribute to the project. ... In fact it is a handbook of present day status of computer algebra. It may serve as an encyclopedia of computer algebra for an interested user. Moreover it might be useful for any potential user to consult the book for the presnet day status of the problems he has in mind. So it could be helpful to pursue her or his own work by integrating methods from computer algebra. One should be grateful to the editors to organize this amount of work in order to integrate the international computer algebra community for such an anlysis of the present day state of the field."

P.Schenzel, Zentralblatt für Mathematik 1017.68162, 2003

"In fact it is a handbook of present day status of the computer algebra. It may serve as an encyclopedia of computer algebra for an interested user. ... So it could be helpful to pursue her or his own work by integrating methods from computer algebra. One should be grateful to the editors to organize this amount of work in order to integrate the international computer algebra community for such an analysis of the present day state of the field." (Peter Schenzel, Zentralblatt MATH, Issue 1017, 2003)

"This is a substantially enlarged report originally covering the state and future of computer algebra in Germany. ... Overall, it gives a high-level ... authoritative overview of what is going on and should be of help to people looking for a research topic, software for specific problems or, later on, writing on the history of the subject." (H. Muthsam, Monatshefte für Mathematik, Vol. 143 (2), 2004)

"This encyclopaedic book is a wonderful source of information on computer algebra. ... Although most people who come to this book will probably use it as an encyclopaedia, it is much more than that. Researchers in computer algebra will benefit from the extensive bibliography, and lecturers will find in it a mine of ideas for interesting topics that might be discussed in a course ... . Finally, to all computer algebra aficionados this book will also provide wonderful entertainment for many a rainy day." (S. C. Coutinho, The Mathematical Gazette, Vol. 88 (512), 2004)

"This handbook is very impressive. Written by more than two hundred specialists, this international edition of a previous German report presents the state of the art of a large field of computational mathematics called computer algebra. Both aspects of this wide domain are well described. ... So, as Bob Caviness says in the preface of the book, 'Gentle reader, I recommend this volume and all its concepts, symbols, and algorithms to you.'" (Jean Moulin Ollagnier, Mathematical Reviews, 2004 h)

"A handbook must be considered as well-composed if it addresses 'all' issues of the area and if its individual articles provide rough surveys of their subject and point the reader to the relevant recent literature. From this point of view, this Computer Algebra Handbook is certainly well-composed ... . The team of editors must be congratulated upon succeeding in collecting contributions of more than 200 authors which make a concise and homogeneous volume." (H. Stetter, Internationale Mathematische Nachrichten, Issue 194, 2003)

"This is a handbook on computer algebra that intends to be as complete as possible, taking into account the large amount of available computer software. The book shows the state of research and applications in the last decade of the twentieth century. ... There are over 200 contributing authors ... resulting in a diversity of styles and concepts. ... a useful handbook both for experts and non-experts, and it could have a broader impact thanks to the diversity of multiple authorship." (Paula Bruggen, Bulletin of the Belgian Mathematical Society, 2007)

Spis treści

1 Development, Characterization, Prospects.- 1.1 Historical Remarks.- 1.2 General Characterization.- 1.3 Impact on Education.- 1.4 Impact on Research.- 1.5 Computer Algebra — Today and Tomorrow.- 1.5.1 Today.- 1.5.2 Outlook.- 2 Topics of Computer Algebra.- 2.1 Exact Arithmetic.- 2.1.1 Long Integer Arithmetic.- 2.1.2 Arithmetic with Polynomials, Rational Functions and Power Series.- 2.1.3 Euchd’s Algorithm and Continued Fractions.- 2.1.4 Modular Arithmetic and the Chinese Remainder Theorem.- 2.1.5 Computations with Algebraic Numbers.- 2.1.6 Real Algebraic Numbers.- 2.1.7 p-adic Numbers and Approximations.- 2.1.8 Finite Fields.- 2.2 Algorithms for Polynomials and Power Series.- 2.2.1 The Division Algorithm.- 2.2.2 Factorization of Polynomials.- 2.2.3 Absolute Factorization of Polynomials.- 2.2.4 Polynomial Decomposition.- 2.2.5 Gröbner Bases.- 2.2.6 Standard Bases.- 2.2.7 Characteristic Sets.- 2.2.8 Algorithmic Invariant Theory.- 2.3 Linear Algebra.- 2.3.1 Linear Systems.- 2.3.2 Algorithms for Matrix CanonicalForms.- 2.4 Constructive Methods of Number Theory.- 2.4.1 Primality Tests.- 2.4.2 Integer Factorization.- 2.4.3 Algebraic Number Fields and AlgebraicFunction Fields.- 2.4.4 Galois Groups.- 2.4.5 Rational Points on Elhptic Curves.- 2.4.6 Geometry of Numbers.- 2.5 Algorithms of Commutative Algebra and Algebraic Geometry.- 2.5.1 Algorithms for Polynomial Ideals and Their Varieties.- 2.5.2 Singularities of Varieties.- 2.5.3 Real Algebraic Geometry.- 2.6 Algorithmic Aspects of the Theory of Algebras.- 2.6.1 Structure Constants.- 2.6.2 Generators and Relations, Swapping and G-Algebras.- 2.6.3 Monad Algebras, Path Algebras and Generalizations.- 2.6.4 Finite-Dimensional Lie Algebras.- 2.6.5 Non-commutative Gröbner Bases.- 2.6.6 Structural Issues and Classification.- 2.6.7 Identities.- 2.6.8 Computational Aspects in the Representation Theory of Quivers and Path Algebras.- 2.7 Computational Group Theory.- 2.7.1 A Crash Course in Group Theory.- 2.7.2 Describing Groups.- 2.7.3 A Brief History.- 2.7.4 Permutation Groups.- 2.7.5 Matrix Groups.- 2.7.6 Black Box Groups.- 2.7.7 Abelian Groups.- 2.7.8 Polycyclic Groups.- 2.7.9 Finitely Presented Groups.- 2.7.10 Group-Theoretic Software.- 2.7.11 Another Perspective.- 2.8 Algorithms of Representation Theory.- 2.8.1 Ordinary Representation Theory.- 2.8.2 Modular Representation Theory.- 2.8.3 Generic Character Tables.- 2.8.4 Summary of Systems.- 2.9 Algebraic Methods for Constructing Discrete Structures.- 2.10 Summation and Integration.- 2.10.1 Definite Summation and Hypergeometric Identities.- 2.10.2 Symbolic Integration.- 2.11 Symbohc Methods for DiflFerential Equations.- 2.11.1 Introduction.- 2.11.2 Differential Galois Theory.- 2.11.3 Lie Symmetries.- 2.11.4 Painlevé Theory.- 2.11.5 Completion.- 2.11.6 Differential Ideal Theory.- 2.11.7 Dynamical Systems.- 2.11.8 Numerical Analysis.- 2.12 Symbolic/Numeric Methods.- 2.12.1 Computer Analysis.- 2.12.2 Algorithms for Computing Validated Results.- 2.12.3 Hybrid Methods.- 2.13 Algebraic Complexity Theory.- 2.14 Coding Theory and Cryptography.- 2.14.1 Coding Theory.- 2.14.2 QuantumCoding Theory.- 2.14.3 Cryptography.- 2.15 Algorithmic Methods inUniversal Algebra and Logic.- 2.15.1 Term Rewriting Systems.- 2.15.2 Decision Procedures and Quantifier Ehmination Methods for AlgebraicTheories.- 2.16 Knowledge Representation and Abstract Data Types.- 2.16.1 Mathematical Knowledge Representation and Expert Systems.- 2.16.2 Abstract Data Types.- 2.17 On the Design of Computer Algebra Systems.- 2.17.1 Memory Management.- 2.17.2 Program Verification and Abstract Data Types.- 2.17.3 The Concept of Types.- 2.17.4 Genericity.- 2.17.5 Modularization.- 2.17.6 Parallel Implementation.- 2.17.7 Continuing Developmentof Computer Algebra Systems.- 2.18 Parahel Computer Algebra Systems.- 2.18.1 Parallel Architectures and Operating Systems Supports.- 2.18.2 Parallel Execution: Mapping and Scheduling.- 2.18.3 Parallelism Expression and Languages.- 2.19 Interfaces and Standardization.- 2.19.1 Interfaces to Word Processors.- 2.19.2 Graphics.- 2.19.3 Interfaces to Numerical Software.- 2.19.4 User Interfaces.- 2.19.5 General Problem-Solving Environments.- 2.19.6 Standardisation.- 2.19.7 MathML.- 2.20 Hardware Implementation of Computer Algebra Algorithms.- 3 Applications of Computer Algebra.- 3.1 Physics.- 3.1.1 Elementary Particle Physics.- 3.1.2 Gravity.- 3.1.3 Central Configurations’in the Newtonian N-Body Problem of Celestial Mechanics.- 3.1.4 CA-Systems for Differential Geometry and Applications.- 3.1.5 Differential Equations in Physics.- 3.2 Mathematics.- 3.2.1 Computer Algebra in Group Theory.- 3.2.2 The Tangent Cone Algorithm and Applications in the Theory of Singularities.- 3.2.3 Automatic Theorem Proving in Geometry.- 3.2.4 Homological Algebra.- 3.2.5 Study of Differential Structures on Quantum Groups.- 3.2.6 Orthogonal Polynomials and Computer Algebra.- 3.2.7 Computer Algebra in Symmetric Bifurcation Theory.- 3.2.8 SymboUc-Numeric Treatment of Equivariant Systems of Equations.- 3.3 Computer Science.- 3.3.1 Computer Algebra in Computer Science.- 3.3.2 Decomposable Structures, Generating Functions and Average-Case of Algorithms.- 3.3.3 Telecommunication Management Networks.- 3.4 Engineering.- 3.4.1 Computer Algebra, a Modern Research Tool for Engineering.- 3.4.2 Critical Load Computations forJet Engines.- 3.4.3 Audio Signal Processing.- 3.4.4 Robotics.- 3.4.5 Computer Aided Design and Modelling.- 3.5 Chemistry.- 3.5.1 Computer Algebra in Chemistry and Crystallography.- 3.5.2 Chemical Reaction Systems.- 3.6 Computer Algebra in Education.- 3.6.1 New Hand-Held ComputerSymbolic Algebra Tools in Mathematics Education.- 3.6.2 The Dutch Perspective.- 3.6.3 Computer Algebra in Teaching and Learning Mathematics: Experiences at the University of Plymouth, England.- 3.6.4 The Educational Use of Computer Algebra Systems at the University of Illinois.- 3.6.5 Mathematics Education from a MATHEMATICA Perspective.- 3.6.6 Visualization: Courseware for Mathematics Education.- 4 Computer Algebra Systems.- 4.1 General Purpose Systems.- 4.1.1 Axiom.- 4.1.2 Aldor.- 4.1.3 Derive and the TI-92.- 4.1.4 Macsyma.- 4.1.5 Magma.- 4.1.6 Maple.- 4.1.7 Mathematica.- 4.1.8 MuPAD.- 4.1.9 Reduce.- 4.2 Special Purpose Systems.- 4.2.1 Algebraic Combinatorics Environment (ACE).- 4.2.2 Building Nonassociative Algebras With Albert.- 4.2.3 Algeb.- 4.2.4 Amore.- 4.2.5 Bergman.- 4.2.6 Cannes/Parcan.- 4.2.7 Carat.- 4.2.8 Casa.- 4.2.9 Chevie.- 4.2.10 C-Meataxe.- 4.2.11 CoCoA.- 4.2.12 Crep.- 4.2.13 The Desir Project and Its Continuation.- 4.2.14 Discreta: A Tool for Constructing i-Designs.- 4.2.15 Felix.- 4.2.16 Fermat.- 4.2.17 FoxBox and Other Blackbox Systems.- 4.2.18 Gap.- 4.2.19 GiNaC.- 4.2.20 Kan/sml.- 4.2.21 Kant V4.- 4.2.22 Lidia.- 4.2.23 Lie.- 4.2.24 Lie.- 4.2.25 A Brief Introduction to Macaulay 2.- 4.2.26 Mas.- 4.2.27 Masyca.- 4.2.28 Moc.- 4.2.29 NTL: A Library for Doing Number Theory.- 4.2.30 Pari.- 4.2.31 Parsac.- 4.2.32 Quotpic.- 4.2.33 Redux.- 4.2.34 Reptiles A Program for Interactively Generating Periodic Tihngs.- 4.2.35 SAC-1, Aldes/SAC-2, Saclib.- 4.2.36 SciNapse: Software that Writes PDE Software.- 4.2.37 Senac.- 4.2.38 Simath — Algorithms in Number Theory.- 4.2.39 SINGULAR — A Computer Algebra System for Polynomial Computations.- 4.2.40 SymbMath.- 4.2.41 Symmetrica.- 4.2.42 Theorema: Computation and Deduction in Natural Style.- 4.2.43 Theorist-a User Interface for Symbolic Algebra.- 4.3 Packages.- 4.3.1 ANU Polycyclic Quotient Programs.- 4.3.2 Arep.- 4.3.3 Cali.- 4.3.4 Cln.- 4.3.5 Crack, Liepde, Applysym and Conlaw.- 4.3.6 Dimsym.- 4.3.7 EinS.- 4.3.8 FeynArts and FormCalc.- 4.3.9 FeynCalc — Tools and Tables for Elementary Particle Physics.- 4.3.10 Grape.- 4.3.11 Recognising Matrix Groups over Finite Fields.- 4.3.12 Molgen.- 4.3.13 Orme.- 4.3.14 Ratappr.- 4.3.15 TTC: Tools of Tensor Calculus.- 5 Meetings and Publications.- 5.1 Conferences and Proceedings.- 5.2 Books on Computer Algebra.- Cited References.- Index for Authors’ Contributions.

O autorze

Prof. Dr. Johannes Grabmeier, Dipl.-Mathematiker, ist Professor für Wirtschaftsinformatik und Informatik an der Fachhochschule Deggendorf. Er lehrt Mathematik, Statistik, Operations Research, Objektorientierte Programmiertechniken, CRM und Data Mining.