Numerische MathematikApproximation Theory

# Lehre

Sommersemester 2010
Title:
Approximation Theory and Approximation Practice
Professor:

Prof. L. N. Trefethen FRS
Date:
Wednesday 14:00 - 16:00

Start: 12.05.2010           End: 30.06.2010
Room:
MA 005   MA Mathematikgebäude
Language:
englisch
Content:
This is a BMS Advanced course for students and researchers interested in numerical computation. Familiarity with Matlab is essential. Some familiarity with approximation theory is desirable but not essential.

The course is built on an unusual book being completed by Prof. Trefethen with the title "Approximation Theory and Approximation Practice: A 21st-Century Treatment in the Form of 32 Executable Chebfun M-Files". It aims to teach both old and new ideas of univariate approximation of functions in a fresh and computational way, illustrating everything through the www2.maths.ox.ac.uk/chebfun system. Both theorems and algorithms will be emphasized: for the former, always with reference to their originator whether in 1912 or 2004; for the latter, always in a hands-on and exploratory fashion.

Topics to be treated include:

* Chebyshev points and interpolants
* Chebyshev polynomials and series
* Barycentric interpolation formula
* Weierstrass Approximation Theory
* Analyticity and convergence rates
* The Gibbs phenomenon
* The Runge phenomenon
* Best approximation and the Remez algorithm
* Lebesgue constants
* Clenshaw-Curtis and Gauss quadrature
* Polynomial roots and colleague matrices
* Approximations based on a conformal map
* Rational functions
Materials:
Chebfun User's Guide and Software
.m and .pdf files of the textbook
Key to all Mythologies

Assignments:
Assignment 1: solns1.html solns1.m solns1.pdf
Exercises 1.1, 1.2, 2.1, 2.2, 2.3, 2.5, 3.1, 3.4, 3.7, 3.8 from the book

Assignment 2:
solns2.html solns2.m solns2.pdf
Exercises 4.1, 4.2, 4.3, 4.5, 5.1, 5.2, 5.3, 5.4 from the book

Assignment 3: solns3.html solns3.m solns3.pdf
Exercises 6.1, 6.2, 7.2, 7.3, 8.2, 8.3, 8.4, 8.6

Assignment 4: solns4.html solns4.m solns4.pdf
Exercises 10.1, 10.2, 10.4, 10.5, 10.6, 11.1, 11.3, 11.4

Assignment 5: solns5.html solns5.m solns5.pdf
Exercises 12.2, 12.3, 13.2, 13.3, 13.4, 14.1, 14.7, 14.9, 14.10

Assignment 6: solns6.html solns6.m solns6.pdf
Exercises 15.1, 16.1, 17.1, 17.2, 17.4, 17.5, 17.6

Assignment 7: solns7.html solns7.m solns7.pdf
Exercises  19.1, 19.4, 19.5, 20.1, 20.2, 20.3, 20.4

Information:
Vorlesungsverzeichnis