Computer Algebra and Symbolic Computation: Mathematical Methods

✍ Scribed by Joel S. Cohen

• Publisher: AK Peters, Ltd.
• Year: 2003
• Tongue: English
• Leaves: 470
• Category: Library

✦ Synopsis

Excellent book covering the basics of building your own computer algebra system, together with its partner text: "Computer Algebra and Symbolic Computation: Elementary Algorithms". The author presents the mathematical fundamentals, practical challenges, formulaic solutions, suggested implementations, and examples in a few programming languages appropriate for someone building their own CAS from scratch in very clear prose. The author makes extensive further reading recommendations for each topic. At each stage of the development, the author presents the motivation for each feature of the CAS along with formal basics of the math behind the techniques. The author presents the basic practical challenges faced by the computer and algorithms in working with the math. Together, these make the text appropriate for both those with math experience moving into programming, and programmers moving towards math. The author lists step-by-step techniques from a mathematician's point of view for solving the problems, and then continues to give algorithm listings in an easily-read "Math Pseudo Language", complete with flow charts explaining the more complex algorithms. A formal computational complexity analysis of the algorithms is not included.The author's examples are provided in Mathematica, MuPad, and Maple, but not in C-like languages. The examples, however, are function/structure-based, and are thus clear enough as to be immediately extensible to the reader's language of choice. For an engineer, the examples amount to a practical recipe. The book also contains a CD with the examples & the text of the book as a .pdf. However, this .pdf is only directly accessible to Windows users. Mac users will need to use Windows to run the included Windows app which can access the .pdf and then save it into another .pdf file to use on their Mac. (I've never seen anything like this before! The .pdf files on the CD show up as "Zero KB Symbolic links" on the Mac which don't resolve unless opened in Windows! In the end, I was able to get the .pdf onto my iPhone so I can easily carry the e-book with me when traveling.)The author provides insight into both the mathematical and computational limitations of the algorithms, the problems caused by the limitations, and frequently gives examples for recognizing the limitations or working around them.Finally, the author provides extensive references for further reading for details on particular topics.In all cases, the author's mastery of English prose makes the information easily accessible, which is not at all the case with some other CAS books I've wasted money on... Speaking of price, considering the extreme niche market and the exquisiteness of the text, the combined price of the two books was worth it.To help the potential customer (as a preview of the contents is not available), this book is the one of the two books (which can be effectively read in any order), and covers the basics of CAS's, the included Math Pseudo Language (MPL), detailed rational number manipulations, automatic simplification of expressions, single and multi-variable polynomial decompositions, resultants, and factorization.It's possible I may one day find a more extensive text with more advanced techniques, but the two texts provide a foundation for building my own CAS.

✦ Table of Contents

Index......Page 1
1.1 Computer Algebra Systems......Page 21
1.2 Mathematical Pseudo-Language (MPL)......Page 22
1.3 Automatic Simplification and Expression Structure......Page 25
1.4 General Polynomial Expressions......Page 31
1.5 Miscellaneous Operators......Page 32
2.1 The Integers......Page 37
2.2 Rational Number Arithmetic......Page 57
2.3 Fields......Page 64
3.1 The Goal of Automatic Simplification......Page 83
3.2 An Automatic Simplification Algorithm......Page 111
4.1 Elementary Concepts and Polynomial Division......Page 131
4.2 Greatest Common Divisors in F[x]......Page 146
4.3 Computations in Elementary Algebraic Number Fields......Page 166
4.4 Partial Fraction Expansion in F(x)......Page 186
5. Polynomial Decomposition......Page 199
5.1 Theoretical Background......Page 200
5.2 A Decomposition Algorithm......Page 208
6.1 Multivariate Polynomials and Integral Domains......Page 221
6.2 Polynomial Division and Expansion......Page 227
6.3 Greatest Common Divisors......Page 249
7.1 The Resultant Concept......Page 285
7.2 Polynomial Relations for Explicit Algebraic Numbers......Page 309
8.1 Multiple Division and Reduction......Page 317
8.2 Equivalence, Simplification, and Ideals......Page 338
8.3 A Simplification Algorithm......Page 354
9. Polynomial Factorization......Page 369
9.1 Square-Free Polynomials and Factorization......Page 370
9.2 Irreducible Factorization: The Classical Approach......Page 380
9.3 Factorization in Zp[x]......Page 390
9.4 Irreducible Factorization: A Modern Approach......Page 419