MA2001 prosjekt nr 3 - våren 2006

Highly oscillatory quadrature (HOS)

In this work we study the integral

\(I(f) = \int_0^1 f(x) \mathrm{exp}(I\omega g(x)) \, \mathrm{d}x\)

Here \(f\) is a given function and \(\omega\)is a real parameter and \(g\) is a given function. This kind of problem appears in several applications, for example in NMR connection. The problem has been dealt with in a wrong way up to now. During my year in Cambridge from 2003 to 2004 we found the right way of dealing with such problems. In this project we shall focus on the following points:

  • Find the value of \(I(f)\) for \(f=\mathrm{exp}(x)\) and \(\omega\) between 0 and 100 by using Gauss quadrature of order 2, 4, 10 and 20 for \(g(x)=x\).
  • Discuss where \(I(f)\) appears in practice by reading the paper AISPN of Iserles and Nørsett (available from Nørsett) and give a list of earlier work on HOS.
  • Do the same experiments as in AISPN.
Veileder: Syvert Nørsett
Rom:  1356 SBII
Telefon:  (735) 93545
E-post:  norsett (at) math.ntnu.no