Gaussian Quadrature
September 12th, 2008 by WaltNumerical integration is not really a field that you expect to have surprising theorems. Yet, the existence of Gaussian quadrature is in of itself surprising. In the elementary methods that you learn in calculus (such as midpoint rule or Simpson’s rule), you evaluate the function at regularly spaced nodes. A more effective technique is to choose so that you can integrate polynomials up to some degree exactly. The best choice? The roots of an orthogonal polynomial.
The proofs are elementary, and can be found in this note by John Cook.
November 8th, 2008 at 11:50 pm
I’d say that the existence of Gaussian quadrature isn’t nearly as surprising as the unreasonable effectiveness of the Trapezium rule applied to a periodic function. This is not to say that the former isn’t surprising, it’s just that the latter demonstrates that sometimes the simplest methods are the best. One of my favourite things about Gaussian quadrature is its link to the convergence of the Conjugate Gradient method (via that Lanczos algorithm).