- Series
- Analysis Seminar
- Time
- Wednesday, February 22, 2012 - 2:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Prof. Avram Sidi – Tecnion-IIT, Haifa, Israel
- Organizer
- Doron Lubinsky
We discuss some recent generalizations of Euler--Maclaurin expansions for the
trapezoidal rule and of analogous asymptotic expansions for Gauss--Legendre
quadrature, in the presence of arbitrary algebraic-logarithmic endpoint
singularities. In addition of being of interest by themselves, these asymptotic
expansions enable us to design appropriate variable transformations to improve
the accuracies of these quadrature formulas arbitrarily. In general, these
transformations are singular, and their singularities can be adjusted easily to
achieve this improvement. We illustrate this issue with a numerical example
involving Gauss--Legendre quadrature.
We also discuss some recent asymptotic expansions of the coefficients of
Legendre polynomial expansions of functions over a finite interval, assuming
that the functions may have arbitrary algebraic-logarithmic interior and
endpoint suingularities. These asymptotic expansions can be used to make
definitive statements on the convergence acceleration rates of extrapolation
methods as these are applied to the Legendre polynomial expansions.