About the Book
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.
Table of Contents:
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.- 2 Topics of Computer Algebra.- 2.1 Exact Arithmetic.- 2.2 Algorithms for Polynomials and Power Series.- 2.3 Linear Algebra.- 2.4 Constructive Methods of Number Theory.- 2.5 Algorithms of Commutative Algebra and Algebraic Geometry.- 2.6 Algorithmic Aspects of the Theory of Algebras.- 2.7 Computational Group Theory.- 2.8 Algorithms of Representation Theory.- 2.9 Algebraic Methods for Constructing Discrete Structures.- 2.10 Summation and Integration.- 2.11 Symbohc Methods for DiflFerential Equations.- 2.12 Symbolic/Numeric Methods.- 2.13 Algebraic Complexity Theory.- 2.14 Coding Theory and Cryptography.- 2.15 Algorithmic Methods inUniversal Algebra and Logic.- 2.16 Knowledge Representation and Abstract Data Types.- 2.17 On the Design of Computer Algebra Systems.- 2.18 Parahel Computer Algebra Systems.- 2.19 Interfaces and Standardization.- 2.20 Hardware Implementation of Computer Algebra Algorithms.- 3 Applications of Computer Algebra.- 3.1 Physics.- 3.2 Mathematics.- 3.3 Computer Science.- 3.4 Engineering.- 3.5 Chemistry.- 3.6 Computer Algebra in Education.- 4 Computer Algebra Systems.- 4.1 General Purpose Systems.- 4.2 Special Purpose Systems.- 4.3 Packages.- 5 Meetings and Publications.- 5.1 Conferences and Proceedings.- 5.2 Books on Computer Algebra.- Cited References.- Index for Authors’ Contributions.
Review :
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 fur 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 fur 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 fur 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)
