Computer algebra : an algorithm-oriented introduction / Wolfram Koepf.

Format:

Book

Author/Creator:

Koepf, Wolfram, author.

Series:

Springer undergraduate texts in mathematics and technology.

Springer Undergraduate Texts in Mathematics and Technology

Language:

English

Subjects (All):

Algebra--Data processing.

Algebra.

Computer science--Mathematics.

Computer science.

Physical Description:

1 online resource (394 pages)

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

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.

Contents:

Intro

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.

Notes:

Description based on print version record.

ISBN:

3-030-78017-1

OCLC:

1260343578