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E-raamat: Computer Algebra: Concepts and Techniques

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  • Formaat: 372 pages
  • Ilmumisaeg: 15-Jan-2019
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351605830
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Computer Algebra: Concepts and TechniquesSuurem pilt
  • Formaat: 372 pages
  • Ilmumisaeg: 15-Jan-2019
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351605830
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The goal of Computer Algebra: Concepts and Techniques is to demystify computer algebra systems for a wide audience, including students, faculty, and professionals in scientific fields such as computer science, mathematics, engineering, and physics. Unlike previous books, the only prerequisites are knowledge of first year calculus and a little programming experience — a background that can be assumed of the intended audience. The book is written in a lean and lively style, with numerous examples to illustrate the issues and techniques discussed. It presents the principal algorithms and data structures, while also discussing the inherent and practical limitations of these systems.

Arvustused

This is a very clearly written introductory textbook on computer algebra systems (CAS), focussing on some of the subjects commonly-used algorithms. It is aimed at beginners and assumes a good background in college algebra along with first-year calculus. It covers representative algorithms in a variety of areas, and these are done in enough detail that you would be able to implement them yourself if you wanted. [ . . . .] The present book has just enough to get you started in computer algebra, and does a very good job of it.

Allen Stenger, MAA Reviews

Preface xi
Acknowledgments xix
1 Introduction
1(28)
1.1 What is Computer Algebra?
1(3)
1.2 Applications
4(6)
1.2.1 Celestial Mechanics
4(3)
1.2.2 Chemical Stoichiometry
7(1)
1.2.3 Geometry
8(2)
1.3 History
10(12)
1.3.1 Early Systems (1961--1980)
11(5)
1.3.2 More Recent Systems (1981--present)
16(6)
1.4 Exercises
22(7)
2 Computer Algebra Systems
29(52)
2.1 Maple
30(22)
2.1.1 Numbers
30(4)
2.1.2 Polynomials and Rational Functions
34(3)
2.1.3 Variables and Functions
37(2)
2.1.4 Calculus
39(5)
2.1.5 Equation Solving
44(2)
2.1.6 Data Structures and Programming
46(6)
2.2 Mathematica
52(20)
2.2.1 Numbers
52(3)
2.2.2 Polynomials and Rational Functions
55(2)
2.2.3 Variables and Functions
57(2)
2.2.4 Calculus
59(4)
2.2.5 Equation Solving
63(2)
2.2.6 Lists
65(3)
2.2.7 Programming
68(4)
2.3 Exercises
72(9)
3 Big Number Arithmetic
81(34)
3.1 Representation of Numbers
82(2)
3.2 Basic Arithmetic
84(10)
3.2.1 Addition and Subtraction
85(1)
3.2.2 Multiplication
86(1)
3.2.3 Karatsuba's Algorithm
87(2)
3.2.4 Division
89(5)
3.3 Greatest Common Divisor
94(9)
3.3.1 Euclid's Algorithm
94(2)
3.3.2 Analysis of Euclid's Algorithm
96(2)
3.3.3 Extended Euclidean Algorithm
98(3)
3.3.4 Binary GCD Algorithm
101(2)
3.4 Rational Arithmetic
103(2)
3.5 Exponentiation
105(4)
3.6 Exercises
109(6)
4 Polynomial Manipulation
115(34)
4.1 Arithmetic on Polynomials
116(11)
4.1.1 Dense and Sparse Representations
116(1)
4.1.2 Addition and Multiplication
117(2)
4.1.3 Karatsuba's Algorithm
119(2)
4.1.4 Sparse Polynomial Multiplication
121(2)
4.1.5 Division
123(1)
4.1.6 Pseudo-division
124(3)
4.2 Polynomial GCDs
127(10)
4.2.1 Adapting Euclid's Algorithm
127(5)
4.2.2 Polynomial Remainder Sequences
132(4)
4.2.3 Extended Euclidean Algorithm
136(1)
4.3 Multivariate Polynomials
137(7)
4.3.1 Expanded and Recursive Forms
138(1)
4.3.2 Data Structures for Polynomials
138(4)
4.3.3 Multivariate GCD Computation
142(2)
4.4 Exercises
144(5)
5 Algebraic Simplification
149(44)
5.1 Issues in Simplification
150(3)
5.2 Normal and Canonical Forms
153(4)
5.3 Simplification of Radicals
157(12)
5.3.1 Algebraic Numbers and Functions
157(4)
5.3.2 Rationalizing Denominators
161(3)
5.3.3 Denesting Radicals
164(5)
5.4 Transcendental Functions
169(18)
5.4.1 Brown's Normal Form Algorithm
170(3)
5.4.2 Structure Theorem
173(4)
5.4.3 Expanded and Combined Forms
177(3)
5.4.4 Euler's Formula
180(3)
5.4.5 Trigonometric Functions
183(4)
5.5 Exercises
187(6)
6 Factorization
193(32)
6.1 What is Factoring?
193(2)
6.2 Rational Roots and Linear Factors
195(1)
6.3 Schubert-Kronecker Factorization
196(3)
6.4 Simplifying Polynomial Factorization
199(5)
6.4.1 Monic Polynomials
199(1)
6.4.2 Square-Free Decomposition
200(4)
6.5 Roundabout Factorization
204(6)
6.5.1 Factoring Polynomials mod m
204(3)
6.5.2 Choosing a Modulus m
207(2)
6.5.3 Roundabout Factorization Algorithm
209(1)
6.6 Distinct Degree Factorization
210(6)
6.6.1 Finding Distinct Degree Factors
210(2)
6.6.2 Splitting Equal Degree Factors
212(4)
6.7 Hensel Lifting
216(4)
6.8 Exercises
220(5)
7 Symbolic Integration
225(44)
7.1 Antidifferentiation
226(2)
7.2 Heuristic Methods
228(4)
7.2.1 Derivative-Divides Method
229(1)
7.2.2 Integration by Parts
230(1)
7.2.3 Elementary Transformations
231(1)
7.3 Rational Function Integration
232(3)
7.4 Hermite's Method
235(5)
7.5 Horowitz' Method
240(3)
7.6 Rothstein-Trager Method
243(6)
7.7 Liouville's Principle
249(2)
7.8 Risch Algorithm: Logarithmic Case
251(7)
7.8.1 Polynomial Part
251(4)
7.8.2 Logarithmic Part
255(2)
7.8.3 Rational Part
257(1)
7.9 Risch Algorithm: Exponential Case
258(4)
7.9.1 Polynomials of Exponentials
258(3)
7.9.2 Rational Functions of Exponentials
261(1)
7.10 Exercises
262(7)
8 Grobner Bases
269(28)
8.1 Solution of Polynomial Equations
270(2)
8.2 Mathematical Applications
272(5)
8.2.1 Linear Algebra
272(2)
8.2.2 Greatest Common Divisors
274(1)
8.2.3 Implicitization of Parametric Equations
275(1)
8.2.4 Integer Programming
276(1)
8.3 General Polynomial Division
277(5)
8.3.1 Ordering of Monomials
277(1)
8.3.2 Issues in Polynomial Division
278(4)
8.4 Construction of Grobner Bases
282(7)
8.4.1 S-polynomials
282(2)
8.4.2 An Example
284(1)
8.4.3 Buchberger's Algorithm
285(1)
8.4.4 Reduced Grobner Bases
286(1)
8.4.5 Efficiency Considerations
287(2)
8.5 Exercises
289(8)
9 Mathematical Correctness
297(34)
9.1 Software Bugs
298(1)
9.2 Limitations of Computer Algebra
299(1)
9.3 Sets of Measure Zero
300(2)
9.3.1 Cancellation of Polynomial GCDs
300(1)
9.3.2 Integrating ∫ xk dx
301(1)
9.4 Indeterminate Forms
302(1)
9.5 Spurious Solutions
303(7)
9.5.1 Extraneous Roots
303(1)
9.5.2 Improper Integrals
304(1)
9.5.3 Discontinuities in Antiderivatives
305(5)
9.6 Branch Cuts
310(6)
9.6.1 Fractional Powers
313(1)
9.6.2 Logarithms
314(1)
9.6.3 Trigonometric Functions
315(1)
9.7 Solution of Equations
316(9)
9.7.1 Linear and Polynomial Equations
317(1)
9.7.2 Equations with Algebraic Functions
318(2)
9.7.3 Exponentials and Logarithms
320(2)
9.7.4 Trigonometric Equations
322(3)
9.8 Concluding Remarks
325(1)
9.9 Exercises
326(5)
Bibliography 331(10)
Index 341(8)
About the Author 349
Edmund A. Lamagna is a professor of computer science at the University of Rhode Island. His professional interests lie at the intersection of computer science and mathematics. In particular, he has contributed to the fields of computer algebra and to the design and analysis of algorithms. In recent years, Ed has become interested in technical and societal aspects of cybersecuity and personal privacy. Throughout his career, he has been involved in the development of innovative approaches for teaching and learning mathematics and computer science.
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