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

• Record Nr.: UNINA9910492138303321
• Autore: Koepf Wolfram
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-78017-1
• Descrizione fisica: 1 online resource (394 pages)
• Collana: Springer Undergraduate Texts in Mathematics and Technology
• Disciplina: 512.0285
• Soggetti: Algebra - Data processing; Computer science - Mathematics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: 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.