Lecture 28.
Adaptive quadrature. Gaussian
quadrature.
Today I finished the presentation I
started last week on adaptive quadrature. Adaptive quadrature is a technique
that automatically determines how many panels to use (and where to put them) so
that a numerical integral is computed within the prescribed accuracy. We saw how
the rounding error could be estimated, and how this estimate is used in a
recursive adaptive Simpson's rule algorithm. I wrote out an algorithm for
recursive adaptive Simpson's rule
quadrature.
We then moved on to
a different approach to numerical quadrature, Gaussian Quadrature. We began by
looking at integrating a degree polynomial in the interval -1 .. 1. We evaluated
the integral by hand and used the integral to derive four constants,
c1,c2, x1, x2, such that the value
of the integral could be obtained by the weighted sum c1*f(x1) + c2*f(x2).
I will continue with Gaussian
quadrature on Thursday and we will see how it can be used to integrate in a more
general setting.
Posted: Tue - November 15, 2005 at 12:09 PM