Computer Algebra: An Algorithm-Oriented Introduction
- Length: 396 pages
- Edition: 1
- Language: English
- Publisher: Springer
- Publication Date: 2021-08-12
- ISBN-10: 3030780163
- ISBN-13: 9783030780166
- Sales Rank: #0 (See Top 100 Books)
This textbook offers an algorithmic introduction to the field of computer algebra. A leading expert in the field, the author guides readers through numerous hands-on tutorials designed to build practical skills and algorithmic thinking. This implementation-oriented approach equips readers with versatile tools that can be used to enhance studies in mathematical theory, applications, or teaching. Presented using Mathematica code, the book is fully supported by downloadable sessions in Mathematica, Maple, and Maxima.
Opening with an introduction to computer algebra systems and the basics of programming mathematical algorithms, the book goes on to explore integer arithmetic. A chapter on modular arithmetic completes the number-theoretic foundations, which are then applied to coding theory and cryptography. From here, the focus shifts to polynomial arithmetic and algebraic numbers, with modern algorithms allowing the efficient factorization of polynomials. The final chapters offer extensions into more advanced topics: simplification and normal forms, power series, summation formulas, and integration.
Computer Algebra is an indispensable resource for mathematics and computer science students new to the field. Numerous examples illustrate algorithms and their implementation throughout, with online support materials to encourage hands-on exploration. Prerequisites are minimal, with only a knowledge of calculus and linear algebra assumed. In addition to classroom use, the elementary approach and detailed index make this book an ideal reference for algorithms in computer algebra.
Preface Contents Chapter 1 Introduction to Computer Algebra 1.1 Capabilities of Computer Algebra Systems 1.2 Additional Remarks 1.3 Exercises Chapter 2 Programming in Computer Algebra Systems 2.1 Internal Representation of Expressions 2.2 Pattern Matching 2.3 Control Structures 2.4 Recursion and Iteration 2.5 Remember Programming 2.6 Divide-and-Conquer Programming 2.7 Programming through Pattern Matching 2.8 Additional Remarks 2.9 Exercises Chapter 3 Number Systems and Integer Arithmetic 3.1 Number Systems 3.2 Integer Arithmetic: Addition and Multiplication 3.3 Integer Arithmetic: Division with Remainder 3.4 The Extended Euclidean Algorithm 3.5 Unique Factorization 3.6 Rational Arithmetic 3.7 Additional Remarks 3.8 Exercises Chapter 4 Modular Arithmetic 4.1 Residue Class Rings 4.2 Modulare Square Roots 4.3 Chinese Remainder Theorem 4.4 Fermat’s Little Theorem 4.5 Modular Logarithms 4.6 Pseudoprimes 4.7 Additional Remarks 4.8 Exercises Chapter 5 Coding Theory and Cryptography 5.1 Basic Concepts of Coding Theory 5.2 Prefix Codes 5.3 Check Digit Systems 5.4 Error Correcting Codes 5.5 Asymmetric Ciphers 5.6 Additional Remarks 5.7 Exercises Chapter 6 Polynomial Arithmetic 6.1 Polynomial Rings 6.2 Multiplication: The Karatsuba Algorithm 6.3 Fast Multiplication with FFT 6.4 Division with Remainder 6.5 Polynomial Interpolation 6.6 The Extended Euclidean Algorithm 6.7 Unique Factorization 6.8 Squarefree Factorization 6.9 Rational Functions 6.10 Additional Remarks 6.11 Exercises Chapter 7 Algebraic Numbers 7.1 Polynomial Quotient Rings 7.2 Chinese Remainder Theorem 7.3 Algebraic Numbers 7.4 Finite Fields 7.5 Resultants 7.6 Polynomial Systems of Equations 7.7 Additional Remarks 7.8 Exercises Chapter 8 Factorization in Polynomial Rings 8.1 Preliminary Considerations 8.2 Efficient Factorization in Zp[x] 8.3 Squarefree Factorization of Polynomials over Finite Fields 8.4 Efficient Factorization in Q[x] 8.5 Hensel Lifting 8.6 Multivariate Factorization 8.7 Additional Remarks 8.8 Exercises Chapter 9 Simplification and Normal Forms 9.1 Normal Forms and Canonical Forms 9.2 Normal Forms and Canonical Forms for Polynomials 9.3 Normal Forms for Rational Functions 9.4 Normal Forms for Trigonometric Polynomials 9.5 Additional Remarks 9.6 Exercises Chapter 10 Power Series 10.1 Formal Power Series 10.2 Taylor Polynomials 10.3 Computation of Formal Power Series 10.3.1 Holonomic Differential Equations 10.3.2 Holonomic Recurrence Equations 10.3.3 Hypergeometric Functions 10.3.4 Efficient Computation of Taylor Polynomials of Holonomic Functions 10.4 Algebraic Functions 10.5 Implicit Functions 10.6 Additional Remarks 10.7 Exercises Chapter 11 Algorithmic Summation 11.1 Definite Summation 11.2 Difference Calculus 11.3 Indefinite Summation 11.4 Indefinite Summation of Hypergeometric Terms 11.5 Definite Summation of Hypergeometric Terms 11.6 Additional Remarks 11.7 Exercises Chapter 12 Algorithmic Integration 12.1 The Bernoulli Algorithm for Rational Functions 12.2 Algebraic Prerequisites 12.3 Rational Part 12.4 Logarithmic Case 12.5 Additional Remarks 12.6 Exercises References List of Symbols Mathematica List of Keywords Index
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