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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000001947682 | QA155.7.E4 G43 1992 | Open Access Book | Book | Searching... |
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Summary
Summary
Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.
Reviews 1
Choice Review
Although research advances in mathematics generally lie too deep to enter the undergraduate curriculum as anything more than footnotes, the profound impact of symbolic manipulation programs such as Axiom, Derive, MACSYMA, Maple, Mathematica, and REDUCE on mathematical practice generally is starting a revolution in the teaching of abstract algebra. Even in traditional courses, students will benefit from working examples of a depth and complexity unapproachable without a computer. Eventually the itinerary and emphasis of such courses will veer towards those topics best illustrated by computer. The most profound change will come when the mathematics that underlies these programs enters the undergraduate classroom. Though barely two decades old, this mathematics is actually close in spirit to the elementary algebra that students meet in high school, in a way that group theory is not. Gr"obner Bases (GB) in particular deserves comparison with David Cox, John Little, and Donal O'Shea's Ideals, Varieties, and Algorithms (CH, May'93) (IVA). Its formidable title notwithstanding, GB is really a very leisurely, general introduction to the algebra of polynomial rings and assumes very little in the way of background from the reader. Gr"obner bases, a tool for determining which polynomials belong to a given ideal in a polynomial ring, first appear after nearly 200 pages of preparatory material, as compared with less than 50 in IVA. Where GB is thoroughgoingly algebraic, IVA is more geometrical. Both books play off structural results against computational procedures. Where GB is quite focused, IVA contains diverse mathematical and even physical applications. The exposition in GB is very complete and exercises are scarce there; exercises form an important part of IVA. The authors of GB suggest that their algorithms be implemented for practice on the reader's system of choice (and recommend the public domain program MAS), but a reading of GB, unlike IVA, does not actually require a computer at hand. Although they cover similar ground, GB and IVA are temperamentally quite distinct, and should be regarded as complementing one another. The authors of Algorithms for Computer Algebra (ACA) have been involved in the design and implementation of Maple. Though ACA devotes one chapter to Gr"obner bases, it is quite unlike GB and IVA. Most of ACA is devoted to practical algorithms and delicate computational issues surrounding conceptually elementary mathematics, such as multiplying polynomials, calculating polynomial greatest common divisors, factoring polynomials, and the like. Unlike GB and IVA, ACA does not pursue structural results for their own sake, but rather emphasizes algorithm analysis, data structures, and implementation details generally lacking in the other books. ACA, alone of the three, treats symbolic integration. Both books under review are highly recommended. Undergraduate. D. V. Feldman; University of New Hampshire
