Algorithms for Computer Algebra

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Springer Science & Business Media, 2007. jún. 30. - 586 oldal
Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter.
Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.
 

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Tartalomjegyzék

Introduction to Computer Algebra
1
12 Symbolic versus Numeric Computation
2
13 A Brief Historical Sketch
4
MAPLE
11
Exercises
20
Algebra Polynomials Rational Functions and Power Series
23
23 Divisibility and Factorization in Integral Domains
26
24 The Euclidean Algorithm
32
Polynomial GCD Computation
279
72 Polynomial Remainder Sequences
280
73 The Sylvester Matrix and Subresultants
285
74 The Modular GCD Algorithm
300
75 The Sparse Modular GCD
311
The EZGCD Algorithm
314
77 A Heuristic Polynomial GCD Algorithm
320
Exercises
331

25 Univariate Polynomial Domains
38
26 Multivariate Polynomial Domains
46
27 The Primitive Euclidean Algorithm
52
28 Quotient Fields and Rational Functions
60
29 Power Series and Extended Power Series
63
210 Relationships among Domains
70
Exercises
73
Normal Forms and Algebraic Representation
79
33 Normal Form and Canonical Form
80
34 Normal Forms for Polynomials
84
35 Normal Forms for Rational Functions and Power Series
88
36 Data Structures for Multiprecision Integers and Rational Numbers
93
37 Data Structures for Polynomials Rational Functions and Power Series
96
Exercises
105
Arithmetic of Polynomials Rational Functions and Power Series
110
42 Basic Arithmetic Algorithms
112
Karatsubas Algorithm
118
44 Modular Representations
120
45 The Fast Fourier Transform
123
46 The Inverse Fourier Transform
128
47 Fast Polynomial Multiplication
132
48 Computing Primitive Nth Roots of Unity
133
49 Newtons Iteration for Power Series Division
136
Exercises
145
Homomorphisms and Chinese Remainder Algorithms
151
53 Ring Morphisms
153
54 Characterization of Morphisms
160
55 Homomorphic Images
167
56 The Integer Chinese Remainder Algorithm
174
57 The Polynomial Interpolation Algorithm
183
58 Further Discussion of the Two Algorithms
189
Exercises
196
Newtons Iteration and the Hensel Construction
204
63 Newtons Iteration for Fu0
214
64 Hensels Lemma
226
65 The Univariate Hensel Lifting Algorithm
232
66 Special Techniques for the Nonmonic Case
240
67 The Multivariate Generalization of Hensels Lemma
250
68 The Multivariate Hensel Lifting Algorithm
260
Exercises
274
Polynomial Factorization
336
83 SquareFree Factorization Over Finite Fields
343
84 Berlekamps Factorization Algorithm
347
85 The Big Prime Berlekamp Algorithm
359
86 Distinct Degree Factorization
368
87 Factoring Polynomials over the Rationals
374
88 Factoring Polynomials over Algebraic Number Fields
378
Exercises
384
Solving Systems of Equations
389
92 Linear Equations and Gaussian Elimination
390
93 FractionFree Gaussian Elimination
393
94 Alternative Methods for Solving Linear Equations
399
95 Nonlinear Equations and Resultants
405
Exercises
422
Gröbner Bases for Polynomial Ideals
429
102 Term Orderings and Reduction
431
103 Gröbner Bases and Buchbergers Algorithm
439
104 Improving Buchbergers Algorithm
447
105 Applications of Gröbner Bases
451
106 Additional Applications
462
Exercises
466
Integration of Rational Functions
473
112 Basic Concepts of Differential Algebra
474
Hermites Method
482
Horowitz Method
488
115 Logarithmic Part of the Integral
492
Exercises
508
The Risch Integration Algorithm
511
122 Elementary Functions
512
123 Differentiation of Elementary Functions
519
124 Liouvilles Principle
523
125 The Risch Algorithm for Transcendental Elementary Functions
529
126 The Risch Algorithm for Logarithmic Extensions
530
127 The Risch Algorithm for Exponential Extensions
547
128 Integration of Algebraic Functions
561
Exercises
569
Notation
574
Index
577
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