Modular Algorithms in Symbolic Summation and Symbolic Integration

  • Jürgen Gerhard

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3218)

Table of contents

  1. Front Matter
  2. Jürgen Gerhard
    Pages 1-5
  3. Jürgen Gerhard
    Pages 7-25
  4. Jürgen Gerhard
    Pages 27-40
  5. Jürgen Gerhard
    Pages 41-60
  6. Jürgen Gerhard
    Pages 79-95
  7. Back Matter

About this book

Introduction

This work brings together two streams in computer algebra: symbolic integration and summation on the one hand, and fast algorithmics on the other hand. In many algorithmically oriented areas of computer science, theanalysisof- gorithms–placedintothe limelightbyDonKnuth’stalkat the 1970ICM –provides a crystal-clear criterion for success. The researcher who designs an algorithmthat is faster (asymptotically, in the worst case) than any previous method receives instant grati?cation: her result will be recognized as valuable. Alas, the downside is that such results come along quite infrequently, despite our best efforts. An alternative evaluation method is to run a new algorithm on examples; this has its obvious problems, but is sometimes the best we can do. George Collins, one of the fathers of computer algebra and a great experimenter,wrote in 1969: “I think this demonstrates again that a simple analysis is often more revealing than a ream of empirical data (although both are important). ” Within computer algebra, some areas have traditionally followed the former methodology, notably some parts of polynomial algebra and linear algebra. Other areas, such as polynomial system solving, have not yet been amenable to this - proach. The usual “input size” parameters of computer science seem inadequate, and although some natural “geometric” parameters have been identi?ed (solution dimension, regularity), not all (potential) major progress can be expressed in this framework. Symbolic integration and summation have been in a similar state.

Keywords

Computer algorithmics algorithms analysis of algorithms computer algebra computer science hermite integration hyperexponential integration hypergeometric summation modular algorithms runtime analysis symbolic computation symbolic integration

Authors and affiliations

  • Jürgen Gerhard
    • 1
  1. 1.MaplesoftWaterlooCanada

Bibliographic information

  • DOI https://doi.org/10.1007/b104035
  • Copyright Information Springer-Verlag Berlin Heidelberg 2005
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Computer Science
  • Print ISBN 978-3-540-24061-7
  • Online ISBN 978-3-540-30137-0
  • Series Print ISSN 0302-9743
  • Series Online ISSN 1611-3349
  • About this book