Modern Computer Algebra (英語) ハードカバー – 2003/9/1
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Computer algebra systems are gaining importance in all areas of science and engineering. This textbook gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. It is designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics. Its comprehensiveness and authority also make 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; a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). Some of this material has never appeared before in book form. For the new edition, errors have been corrected, the text has been smoothed and updated, and new sections on greatest common divisors and symbolic integration have been added.
From reviews of the first edition: 'Wow! What a beautifully produced book, and what a wealth of information.' Don Knuth
'This book is a delight: I heartily recommend it.' London Mathematical Society Newsletter
'I predict it will be a major success.' Steve Cook, University of Toronto
'I find the quality of this book really exceptional …'. Zentralblatt für Mathematik
'… this lively and exciting volume represents the state of the art in textbooks on computer algebra. Every student and instructor in this area will want a copy.' Mathematical Reviews
Since Amazon doesn't sell this except through third parties, you have to be very careful you don't get the 1999 edition by accident. In fact, many sellers don't even check or know there is an update. I suggest you email the seller you choose here on Amazon and be sure they check that it's the 2003 update before they ship to save you the hassle of returns. There is an ISBN difference (The 2003 ISBN (10) is 0521826462, whereas the 1999 edition is 0-521-64176-4). This is not really a new EDITION, as it is an update, which further confuses the issue, because you can't just look at the cover.
Symbolic integration and greatest common divisors were added in 2003. Both computer arithmetic and algebra are covered at a relatively deep level. I'd suggest that if you're new to CAS, you start with this text, then move on to the more detailed specialty books I've listed below. When TI moved from their earlier calculators to the TI-83/84 +, then the TI-89, Voyage 200 and finally the recent Nspire (89 with QWERTY), symbolic integration not only of algebra, but also ODEs and even PDE's became the norm. Of course Maple, the desktop version of Nspire, etc. also use the base assembly language wired into the handhelds when they access your PC's APU/GPU and CPU.
Here are a few of the best more recent books that augment Gathen and Gerhard's fine text:
Both of Cohen's books: Computer Algebra and Symbolic Computation: Mathematical Methods and Computer Algebra and Symbolic Computation: Elementary Algorithms
Muller: Elementary Functions: Algorithms and Implementation
Lu: Arithmetic and Logic in Computer Systems (Wiley Series in Microwave and Optical Engineering)
Koren: Computer Arithmetic Algorithms, Second Edition
Brent: Modern Computer Arithmetic (Cambridge Monographs on Applied and Computational Mathematics)
See my Listmania list on CAS and Computer Arithmetic for many more, including some that deal with hardware implementation of CAS circuits with FPGA, synthesis, etc. If you are in High School or have very little math or computer background, I recommend you start with Maxfield Brown's two books first: Bebop to the Boolean Boogie, Third Edition: An Unconventional Guide to Electronics and The Definitive Guide to How Computers Do Math : Featuring the Virtual DIY Calculator.
After Brown, as you get into undergrad and grad, MCA is still the go to text for comprehensive coverage of the major functions and algorithms. Publishers note: Hey, guys, how about a comprehensive 2013 update???
This is the best treatment I've seen for lattice basis reduction (a topic treated in many other texts) as well as for polynomial factoring over fields (Cantor Zassenhaus) and other topics. I've read parts of this book that were irrelevant for my research simply because the book is a joy to read. Highly recommended to anyone with an interest and some degree of mathematical maturity.
The only drawback is that my edition has a number of typos, some serious (however the authors do provide an online corrigenda).