Lecture 25. 


Numerical Quadrature. 

Today we began our exploration of numerical integration, also known as numerical quadrature. The midpoint, trapezoid and Simpson's methods are three examples of a collection of rules known as Newton-Cotes formulas.

I defined the rules, and then we used them to integrate the function x3. The outcome was rather surprising as the error for the one point rule was smaller than that for the two point rule. Furthermore, Simpson's rule using a quadratic interpolant integrated a cubic polynomial with zero error. This is not just a cooked up example but a property of Simpson's rule.

The error bounds for the Newton-Cotes rules are derived by integrating a Taylor expansion. I tried to give some intuition as why Simpson's rule is perfectly accurate up to cubic polynomials by showing how the odd terms in the integral of the Taylor polynomial vanish.

On Thursday we will look at the composite rules.
 

Posted: Tue - November 8, 2005 at 12:02 PM          

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