Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Now in its third edition, this highly successful textbook is widely regarded as the 'bible of computer algebra'.

Abstract: "Is the use of computer algebra technology beneficial for mechanised reasoning in and about mathematical domains? Usually it is assumed that it is. Many works in this area, however, either have little reasoning content, or use symbolic computation only to simplify expressions. In work that has achieved more, the used methods do not scale up. They trust the computer algebra system either too much or too little. Computer algebra systems are not as rigorous as many provers. They are not logically sound reasoning systems, but collections of algorithms. We classify soundness problems that occur in computer algebra systems. While many algorithms and their implementations are perfectly trustworthy, the semantics of symbols is often unclear and leads to errors. On the other hand, more robust approaches to interface external reasoners to provers are not always practical because the mathematical depth of proofs algorithms in computer algebra are based on can be enormous. Our own approach takes both trustworthiness of the overall system and efficiency into account. It relies on using only reliable parts of a computer algebra system, which can be achieved by choosing a suitable library, and deriving specifications for these algorithms from their literature. We design and implement an interface between the prover Isabelle and the computer algebra library Sumit [sic] and use it to prove non-trivial theorems from coding theory. This is based on the mechanisation of the algebraic theories of rings and polynomials. Coding theory is an area where proofs do have a substantial amount of computational content. Also, it is realistic to assume that the verification of an encoding or decoding device could be undertaken in, and indeed, be simplified by, such a system. The reason why semantics of symbols is often unclear in current computer algebra systems is not mathematical difficulty, but the design of those systems. For Gaussian elimination we show how the soundness problem can be fixed by a small extension, and without losing efficiency. This is a prerequisite for the efficient use of the algorithm in a prover."

This is the second of two books on Computer Algebra in Industry whose contents are based on the assertion that the proper use of computer algebra, especially the general purpose software packages presently available, will enhance the capabilities of technical and mathematical researchers in industry. Reflecting the rapid growth in the area, the contributions are written with the non-expert in mind with the intention of demonstrating the effective use and cost effectiveness of computer algebra.

Based on the proceedings of the SCAFI '91 Seminar, this collection of research papers demonstrates the useful and cost-effective applications of computer algebra in industry. The core of the text is a number of case studies of industrial applications by European companies of diverse origins.

"This volume contains the papers, updated in some cases, presented at the first AISMC (Artificial Intelligence and Symbolic Mathematical Computations)conference, held in Karlsruhe, August 3-6, 1992. This was the first conference to be devoted to such a topic after a long period when SMC made no appearance in AI conferences, though it used to be welcome in the early days of AI. Some conferences were held recently on mathematics and AI, but none was directly comparable in scope to this conference. Because of the novelty of the domain, authors were given longer allocations of time than usual in which to present their work. As a result, extended and fruitful discussions followed each paper. The introductory chapter in this book, which was not presented during the conference, reflects in many ways the flavor of these discussions and aims to set out the framework for future activities in this domain of research. In addition to the introduction, the volume contains 20 papers."--PUBLISHER'S WEBSITE.