# Algorithms for Computer Algebra

Springer Science & Business Media, Jun 30, 2007 - Computers - 586 pages
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.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### Contents

 Introduction to Computer Algebra 1 12 Symbolic versus Numeric Computation 2 13 A Brief Historical Sketch 4 MAPLE 11 Exercises 20 Algebra Polynomials Rational Functions and Power Series 23 23 Divisibility and Factorization in Integral Domains 26 24 The Euclidean Algorithm 32
 Polynomial GCD Computation 279 72 Polynomial Remainder Sequences 280 73 The Sylvester Matrix and Subresultants 285 74 The Modular GCD Algorithm 300 75 The Sparse Modular GCD 311 The EZGCD Algorithm 314 77 A Heuristic Polynomial GCD Algorithm 320 Exercises 331

 25 Univariate Polynomial Domains 38 26 Multivariate Polynomial Domains 46 27 The Primitive Euclidean Algorithm 52 28 Quotient Fields and Rational Functions 60 29 Power Series and Extended Power Series 63 210 Relationships among Domains 70 Exercises 73 Normal Forms and Algebraic Representation 79 33 Normal Form and Canonical Form 80 34 Normal Forms for Polynomials 84 35 Normal Forms for Rational Functions and Power Series 88 36 Data Structures for Multiprecision Integers and Rational Numbers 93 37 Data Structures for Polynomials Rational Functions and Power Series 96 Exercises 105 Arithmetic of Polynomials Rational Functions and Power Series 110 42 Basic Arithmetic Algorithms 112 Karatsubas Algorithm 118 44 Modular Representations 120 45 The Fast Fourier Transform 123 46 The Inverse Fourier Transform 128 47 Fast Polynomial Multiplication 132 48 Computing Primitive Nth Roots of Unity 133 49 Newtons Iteration for Power Series Division 136 Exercises 145 Homomorphisms and Chinese Remainder Algorithms 151 53 Ring Morphisms 153 54 Characterization of Morphisms 160 55 Homomorphic Images 167 56 The Integer Chinese Remainder Algorithm 174 57 The Polynomial Interpolation Algorithm 183 58 Further Discussion of the Two Algorithms 189 Exercises 196 Newtons Iteration and the Hensel Construction 204 63 Newtons Iteration for Fu0 214 64 Hensels Lemma 226 65 The Univariate Hensel Lifting Algorithm 232 66 Special Techniques for the Nonmonic Case 240 67 The Multivariate Generalization of Hensels Lemma 250 68 The Multivariate Hensel Lifting Algorithm 260 Exercises 274
 Polynomial Factorization 336 83 SquareFree Factorization Over Finite Fields 343 84 Berlekamps Factorization Algorithm 347 85 The Big Prime Berlekamp Algorithm 359 86 Distinct Degree Factorization 368 87 Factoring Polynomials over the Rationals 374 88 Factoring Polynomials over Algebraic Number Fields 378 Exercises 384 Solving Systems of Equations 389 92 Linear Equations and Gaussian Elimination 390 93 FractionFree Gaussian Elimination 393 94 Alternative Methods for Solving Linear Equations 399 95 Nonlinear Equations and Resultants 405 Exercises 422 Gröbner Bases for Polynomial Ideals 429 102 Term Orderings and Reduction 431 103 Gröbner Bases and Buchbergers Algorithm 439 104 Improving Buchbergers Algorithm 447 105 Applications of Gröbner Bases 451 106 Additional Applications 462 Exercises 466 Integration of Rational Functions 473 112 Basic Concepts of Differential Algebra 474 Hermites Method 482 Horowitz Method 488 115 Logarithmic Part of the Integral 492 Exercises 508 The Risch Integration Algorithm 511 122 Elementary Functions 512 123 Differentiation of Elementary Functions 519 124 Liouvilles Principle 523 125 The Risch Algorithm for Transcendental Elementary Functions 529 126 The Risch Algorithm for Logarithmic Extensions 530 127 The Risch Algorithm for Exponential Extensions 547 128 Integration of Algebraic Functions 561 Exercises 569 Notation 574 Index 577 Copyright

### References to this book

 A Singular Introduction to Commutative AlgebraLimited preview - 2007
 Advanced Topics in Computational Number TheoryHenri CohenLimited preview - 1999
All Book Search results &raquo;