Algorithms for Computer Algebra

By: K. O. Geddes, G. Labahn, S. R. Czapor

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Hardcover | 30 September 1992

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608 Pages

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24.77 x 16.51 x 3.18

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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.

Industry Reviews

`The Computer Algebra community has been waiting for years for this book to appear. ...the book is a masterpiece and can be recommended to everyone interested in the algorithms for computer algebra, either as a reference for further research or just to give the casual user an idea why things work as good (or as bad) as they do in computer algebra packages. ...the recommendation would be clear: Buy this book! As it stands now my only advice is to stay away from this book, because you might be tempted to buy it anyway (at least I was).'
Computer Algebra Nederland Newsletter, June 1993

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Non-Fiction Mathematics Algebra Computing & I.T.Computer Science Image Processing Databases Data Capture & Analysis Business Applications Mathematical & Statistical Software Computer Programming & Software Development Algorithms & Data Structures

Graphical & Digital Media Applications

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Published: 24th March 2013

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Preface
Introduction to Computer Algebra
Symbolic versus Numeric Computation	p. 2
A Brief Historical Sketch	p. 4
An Example of a Computer Algebra System: MAPLE	p. 11
Algebra of Polynomials, Rational Functions, and Power Series
Rings and Fields	p. 23
Divisibility and Factorization in Integral Domains	p. 26
The Euclidean Algorithm	p. 32
Univariate Polynomial Domains	p. 38
Multivariate Polynomial Domains	p. 46
The Primitive Euclidean Algorithm	p. 52
Quotient Fields and Rational Functions	p. 60
Power Series and Extended Power Series	p. 63
Relationships among Domains	p. 70
Normal Forms and Algebraic Representations
Levels of Abstraction	p. 79
Normal Form and Canonical Form	p. 80
Normal Forms for Polynomials	p. 84
Normal Forms for Rational Functions and Power Series	p. 88
Data Structures for Multiprecision Integers and Rational Numbers	p. 93
Data Structures for Polynomials, Rational Functions, and Power Series	p. 96
Arithmetic of Polynomials, Rational Functions, and Power Series
Basic Arithmetic Algorithms	p. 112
Fast Arithmetic Algorithms: Karatsuba's Algorithm	p. 118
Modular Representations	p. 120
The Fast Fourier Transform	p. 123
The Inverse Fourier Transform	p. 128
Fast Polynomial Multiplication	p. 132
Computing Primitive N-th Roots of Unity	p. 133
Newton's Iteration for Power Series Division	p. 136
Homomorphisms and Chinese Remainder Algorithms
Intermediate Expression Swell: An Example	p. 151
Ring Morphisms	p. 153
Characterization of Morphisms	p. 160
Homomorphic Images	p. 167
The Integer Chinese Remainder Algorithm	p. 174
The Polynomial Interpolation Algorithm	p. 183
Further Discussion of the Two Algorithms	p. 189
Newton's Iteration and the Hensel Construction
P-adic and Ideal-adic Representations	p. 205
Newton's Iteration for F(u)=0	p. 214
Hensel's Lemma	p. 226
The Univariate Hensel Lifting Algorithm	p. 232
Special Techniques for the Non-monic Case	p. 240
The Multivariate Generalization of Hensel's Lemma	p. 250
The Multivariate Hensel Lifting Algorithm	p. 260
Polynomial GCD Computation
Polynomial Remainder Sequences	p. 280
The Sylvester Matrix and Subresultants	p. 285
The Modular GCD Algorithm	p. 300
The Sparse Modular GCD Algorithm	p. 311
GCD's using Hensel Lifting: The EZ-GCD Algorithm	p. 314
A Heuristic Polynomial GCD Algorithm	p. 320
Polynomial Factorization
Square-Free Factorization	p. 337
Square-Free Factorization Over Finite Fields	p. 343
Berlekamp's Factorization Algorithm	p. 347
The Big Prime Berlekamp Algorithm	p. 359
Distinct Degree Factorization	p. 368
Factoring Polynomials over the Rationals	p. 374
Factoring Polynomials over Algebraic Number Fields	p. 378
Solving Systems of Equations
Linear Equations and Gaussian Elimination	p. 390
Fraction-Free Gaussian Elimination	p. 393
Alternative Methods for Solving Linear Equations	p. 399
Nonlinear Equations and Resultants	p. 405
Grobner Bases for Polynomial Ideals
Term Orderings and Reduction	p. 431
Grobner Bases and Buchberger's Algorithm	p. 439
Improving Buchberger's Algorithm	p. 447
Applications of Grobner Bases	p. 451
Additional Applications	p. 462
Integration of Rational Functions
Basic Concepts of Differential Algebra	p. 474
Rational Part of the Integral: Hermite's Method	p. 482
Rational Part of the Integral: Horowitz' Method	p. 488
Logarithmic Part of the Integral	p. 492
The Risch Integration Algorithm
Elementary Functions	p. 512
Differentiation of Elementary Functions	p. 519
Liouville's Principle	p. 523
The Risch Algorithm for Transcendental Elementary Functions	p. 529
The Risch Algorithm for Logarithmic Extensions	p. 530
The Risch Algorithm for Exponential Extensions	p. 547
Integration of Algebraic Functions	p. 561
Notation	p. 575
Index	p. 577
Table of Contents provided by Blackwell. All Rights Reserved.

Algorithms for Computer Algebra

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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