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E-raamat: Computer Algebra: An Algorithm-Oriented Introduction

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Computer Algebra: An Algorithm-Oriented IntroductionSuurem pilt
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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.

Arvustused

Strong interplay between the abstract exposition, which includes the relevant theorems as well as their proofs, and the practical utilization of those concepts in Mathematica is certainly a remarkable feature of this textbook. Overall, the book is very well written and the approach to provide examples as actual Mathematica sessions is commendable. (Andreas Maletti, zbMATH 1484.68004, 2022)

1 Introduction To Computer Algebra
1(20)
1.1 Capabilities of Computer Algebra Systems
1(17)
1.2 Additional Remarks
18(1)
1.3 Exercises
18(3)
2 Programming In Computer Algebra Systems
21(18)
2.1 Internal Representation of Expressions
21(1)
2.2 Pattern Matching
22(1)
2.3 Control Structures
23(2)
2.4 Recursion and Iteration
25(3)
2.5 Remember Programming
28(3)
2.6 Divide-and-Conquer Programming
31(1)
2.7 Programming through Pattern Matching
32(2)
2.8 Additional Remarks
34(1)
2.9 Exercises
34(5)
3 Number Systems And Integer Arithmetic
39(26)
3.1 Number Systems
39(1)
3.2 Integer Arithmetic: Addition and Multiplication
40(9)
3.3 Integer Arithmetic: Division with Remainder
49(3)
3.4 The Extended Euclidean Algorithm
52(4)
3.5 Unique Factorization
56(4)
3.6 Rational Arithmetic
60(1)
3.7 Additional Remarks
61(1)
3.8 Exercises
62(3)
4 Modular Arithmetic
65(26)
4.1 Residue Class Rings
65(5)
4.2 Modulare Square Roots
70(2)
4.3 Chinese Remainder Theorem
72(2)
4.4 Fermat's Little Theorem
74(4)
4.5 Modular Logarithms
78(3)
4.6 Pseudoprimes
81(6)
4.7 Additional Remarks
87(1)
4.8 Exercises
88(3)
5 Coding Theory And Cryptography
91(22)
5.1 Basic Concepts of Coding Theory
91(2)
5.2 Prefix Codes
93(5)
5.3 Check Digit Systems
98(1)
5.4 Error Correcting Codes
99(4)
5.5 Asymmetric Ciphers
103(7)
5.6 Additional Remarks
110(1)
5.7 Exercises
111(2)
6 Polynomial Arithmetic
113(38)
6.1 Polynomial Rings
113(4)
6.2 Multiplication: The Karatsuba Algorithm
117(3)
6.3 Fast Multiplication with FFT
120(8)
6.4 Division with Remainder
128(4)
6.5 Polynomial Interpolation
132(2)
6.6 The Extended Euclidean Algorithm
134(3)
6.7 Unique Factorization
137(6)
6.8 Squarefree Factorization
143(4)
6.9 Rational Functions
147(1)
6.10 Additional Remarks
148(1)
6.11 Exercises
149(2)
7 Algebraic Numbers
151(42)
7.1 Polynomial Quotient Rings
151(4)
7.2 Chinese Remainder Theorem
155(2)
7.3 Algebraic Numbers
157(10)
7.4 Finite Fields
167(6)
7.5 Resultants
173(7)
7.6 Polynomial Systems of Equations
180(6)
7.7 Additional Remarks
186(1)
7.8 Exercises
186(7)
8 Factorization In Polynomial Rings
193(28)
8.1 Preliminary Considerations
193(3)
8.2 Efficient Factorization in Zp[ x]
196(7)
8.3 Squarefree Factorization of Polynomials over Finite Fields
203(2)
8.4 Efficient Factorization in Q[ x]
205(5)
8.5 Hensel Lifting
210(5)
8.6 Multivariate Factorization
215(2)
8.7 Additional Remarks
217(1)
8.8 Exercises
218(3)
9 Simplification And Normal Forms
221(14)
9.1 Normal Forms and Canonical Forms
221(3)
9.2 Normal Forms and Canonical Forms for Polynomials
224(2)
9.3 Normal Forms for Rational Functions
226(1)
9.4 Normal Forms for Trigonometric Polynomials
227(4)
9.5 Additional Remarks
231(1)
9.6 Exercises
232(3)
10 Power Series
235(54)
10.1 Formal Power Series
235(6)
10.2 Taylor Polynomials
241(3)
10.3 Compulation of Formal Power Series
244(25)
10.3.1 Holonomic Differential Equations
247(9)
10.3.2 Holonomic Recurrence Equations
256(5)
10.3.3 Hypergeometric Functions
261(7)
10.3.4 Efficient Computation of Taylor Polynomials of Holonomic Functions
268(1)
10.4 Algebraic Functions
269(5)
10.5 Implicit Functions
274(7)
10.6 Additional Remarks
281(1)
10.7 Exercises
281(8)
11 Algorithmic Summation
289(40)
11.1 Definite Summation
289(7)
11.2 Difference Calculus
296(3)
11.3 Indefinite Summation
299(3)
11.4 Indefinite Summation of Hypergeometric Terms
302(12)
11.5 Definite Summation of Hypergeometric Terms
314(10)
11.6 Additional Remarks
324(1)
11.7 Exercises
325(4)
12 Algorithmic Integration
329(32)
12.1 The Bernoulli Algorithm for Rational Functions"
329(1)
12.2 Algebraic Prerequisites
330(5)
12.3 Rational Part
335(6)
12.4 Logarithmic Case
341(17)
12.5 Additional Remarks
358(1)
12.6 Exercises
359(2)
References 361(4)
List of Symbols 365(2)
Mathematica List of Keywords 367(5)
Index 372
Wolfram Koepf is Professor of Mathematics (retired) at the University of Kassel. He is a leading expert in computer algebra. Between 2002 and 2010 he was the Chairman of the Fachgruppe Computeralgebra, the largest computer algebra group worldwide. He has organized numerous conferences and workshops around the world, including the International Symposium on Symbolic and Algebraic Computation (ISSAC) 2010 as general chair, and a cooperation with the African Institute of Mathematical Sciences Cameroon in 2017 and 2018. In addition to his prolific research and organizational activities, he has a strong interest in education and teacher preparation. He is also a member of the executive committee of the German Mathematical Union (DMV).
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