Computer Analysis

Summer 2014

Description - Course Details

Description

The goal of the course is to understand the underlying principles of algorithms typically used in computer algebra systems for computing closed forms of integrals, or for solving differential equations.

The topics include: Differential fields, integration of rational functions, the Risch algorithm, polynomial and rational solutions of ODEs, hyperexponential solutions, series solutions via Frobenius method and Newton polygon, algorithms for algebraic and holonomic functions.

Bachelor or Masters theses topics which are based on the material of this course are available.

Course Details

• Lecturer: Manuel Kauers
• If you are regular JKU student, please register to this course via KUSSS
• Exam/credits mode: will be discussed in the first meeting
• Time/place: Tuesdays, 16:15--17:45, HS14
• First meeting: March 11 (NO LECTURE ON MARCH 04!)
• Prerequisits: Participants are expected to be acquainted with the basic notions from analysis (differentiation, integration, power series expansions, etc.) and linear algebra (finite dimensional vector spaces, solving linear systems of equations, etc.). It might be an advantage, although it is not a formal requirement, to do the courses Computeralgebra and/or Analytische Kombinatorik before this course.
• Lecture notes:
  1. Motivation, Differential Rings, and the Problem(s) of Symbolic Integration
  2. In-Field Integration of Rational Functions: Hermite Reduction
  3. Elementary Integration of Rational Functions: Rothstein-Trager Resultant
  4. Liouville's Theorem
  5. Risch's Algorithm, part I: Hermite Reduction and Rothstein-Trager Resultant
  6. Risch's Algorithm, part II: The Polynomial Part
  7. General remarks about linear ODEs; polynomial solutions
  8. rational solutions; generalized series solutions I (Frobenius method)
  9. generalized series solutions II (Newton polygon); hyperexponential solutions
  10. Picard-Vessiot rings
  11. Differential Galois groups
  12. Closure Properties for D-finite functions
  13. Several variables; definite integration
• Slides with algorithms: algorithms.pdf
• Exam: homework2014.pdf

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