Modular Algorithms in Symbolic Summation and Symbolic Integration 符号求知与符号积分的模块化算法

2005-1  

北京燕山出版社  

Gerhard, Juergen; Gerhard, J]rgen; Gerhard, Jurgen  

224  

This book brings together two streams of computer algebra: symbolic summation and integration on the one hand, and fast algorithmics on the other hand. In symbolic integration and summation, not too many algorithms with analyzed run times are known, and until now the mathematically oriented world of integration and summation and the computer science world of algorithm analysis have not had much to say to each other. The progress presented in this work towards overcoming this situation is threefold: - a clear framework for algorithm analysis with the appropriate parameters is provided, - modular algorithmic techniques are introduced in this area, and - almost optimal algorithms are presented for the basic problems.

1.Introduction2.Overview 2.1 Outline 2.2 Statement of Main Results 2.3 References and Related Works 2.4 Open Problems3.Technical Prerequisites 3.1 Subresultants and the Euclidean Algorithm 3.2 The Cost of Arithmetic4.Change of Basis 4.1 Computing Taylor Shifts 4.2 Conversion to Falling Factorials 4.3 Fast Multiplication in the Falling Factorial Basis5.Modular Squarefree and Greatest Factorial Factorization 5.1 Squarefree Factorization 5.2 Greatest Factorial Factorization6.Modular Hermite Integration 6.1 Small Primes Modular Algorithm 6.2 Prime Power Modular Algorithm 6.3 Implementation7.Computing All Integral Roots of the Resultant 7.1 Application to Hypergeometric Summation 7.2 Computing All Integral Roots Via Factoring 7.3 Application to Hyperexponential Integration 7.4 Modular LRT Algorithm8.Modular Algorithms for the Gosper-Petkov~ek Form 8.1 Modular GP~-Form Computation9.Polynomial Solutions of Linear First Order Equations 9.1 The Method of Undetermined Coefficients 9.2 Brent and Kung's Algorithm for Linear Differential Equations.. 9.3 Rothstein's SPDE Algorithm 9.4 The ABP Algorithm 9.5 A Divide-and-Conquer Algorithm: Generic Case 9.6 A Divide-and-Conquer Algorithm: General Case 9.7 Barkatou's Algorithm for Linear Difference Equations 9.8 Modular Algorithms10. Modular Gosper and Almkvist & Zeilberger Algorithms 10.1 High Degree ExamplesReferencesIndex

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