Computer Algebra and Algebraic Programming - Symbo1ics Ideas

95 Pages·2011·0.58 MB·English

by  Robert Smith

Computer Algebra
and
Algebraic Programming

Computer Algebra
and
Algebraic Programming
Robert Smith
Quad’s Printing — St. Paul
Copyright ©2011 by Robert Smith. All rights reserved.
This is a work-in-progress. This copy was generated on February 23, 2011. If you
happen to find an error, please send an email with relevant details and corrections to
the author at [email protected].
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Information providedin thisbook areprovided“ASIS”and“WITHALLFAULTS”
without any warranty, express or implied. The Authormakes reasonable effort to
includeaccurate and up-to-dateinformation; theAuthordoes not, however, make
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one’s own risk.
TheAuthordoes not warrant or guarantee
1. theaccuracy, adequacy,quality,currentness, validity,completeness, or
suitability of any information for any purpose; or
2. that defects in this book will be corrected.
Contents
Contents v
I The Algebraic Paradigm 1
1 The Symbolic and Algebraic Paradigm 3
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Structure of Algebraic Programs 7
2.1 Symbols, Variables, and Values . . . . . . . . . . . . . . . . . . 7
2.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Mappings and Constructions . . . . . . . . . . . . . . . . . . . 8
2.4 Substitution Semantics . . . . . . . . . . . . . . . . . . . . . . . 8
3 Values and Types 11
3.1 A Discourse on Types . . . . . . . . . . . . . . . . . . . . . . . 11
II Computer Algebra Fundamentals 15
4 Relations, Identities, Forms 17
4.1 Relationships Between Values . . . . . . . . . . . . . . . . . . . 17
4.2 Equality in Computer Algebra . . . . . . . . . . . . . . . . . . 18
4.3 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Normal Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Arbitrary Precision Arithmetic 23
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Computer Representation . . . . . . . . . . . . . . . . . . . . . 24
5.3 High-PrecisionEvaluation of Hypergeometric Series . . . . . . 25
IIIAbstract Algebra 29
6 Basic Structures and Divisibility 31
vi Contents
6.1 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Divisibility and Factorization . . . . . . . . . . . . . . . . . . . 32
7 Polynomial, Rational, and Series Structures 37
7.1 Univariate Polynomial Domains . . . . . . . . . . . . . . . . . . 37
8 Field Extensions 39
8.1 Algebraic and Transcendental Extensions . . . . . . . . . . . . 40
IVPolynomials 43
9 Polynomial Algorithms 45
9.1 Sylvester Matrices and Resultants . . . . . . . . . . . . . . . . 45
9.2 Polynomial Roots and Algebraic Numbers . . . . . . . . . . . . 47
10 Algebra In the Reals 49
10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
10.2 Real Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . 49
V Series 51
11 Hypergeometric Series 53
11.1 Hypergeometric Series and Functions . . . . . . . . . . . . . . . 53
11.2 Series as Hypergeometric Functions . . . . . . . . . . . . . . . . 55
11.3 Discrete Operators and The Summation Problem . . . . . . . . 55
11.4 Gosper’s Algorithm for Indefinite Summation . . . . . . . . . . 57
11.5 Zeilberger’s Algorithm for Definite Summation . . . . . . . . . 60
VIIntegration 63
12 Risch Integration 65
12.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.2 Risch Algorithm: Input . . . . . . . . . . . . . . . . . . . . . . 66
Appendix 71
A Function Tables 71
A.1 Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . 71
A.2 Gamma and Friends . . . . . . . . . . . . . . . . . . . . . . . . 73
A.3 Hypergeometric Identities . . . . . . . . . . . . . . . . . . . . . 75
A.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
B Miscellaneous 77
Contents vii
B.1 The Gudermannian Function . . . . . . . . . . . . . . . . . . . 77
B.2 The Γ-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
B.3 The Lambert W-Function . . . . . . . . . . . . . . . . . . . . . 82
Bibliography 83
Index 85

Part I
The Algebraic Paradigm