The Best Possible Quadrature Routine (Within Reason)

Originally all I wanted to do with numerical analysis was string together some functions and plot an ephemeris, given the right data. Planetary motions are fascinating, and I think it will be a really cool project. However, after reading the first chapter of the textbook Celestial Mechanics, where the author lists important numerical methods, I realized that numerical analysis by itself is something interesting.

In the last section of the first chapter he gives a list of programs or routines that are useful to him. One of those was using Gaussian quadrature to compute a definite integral. I already discussed the problem of quadrature in the post on Simpson's Rule, and I mentioned Gaussian quadrature specifically in the post on Lagrange interpolation. But, it's still worth revisiting the question of what quadrature means, before I share what I learned about Gaussian quadrature.

In calculus we learn that a definite integral is actually the area beneath a curve. We approximate this with a sum of the areas of many small slices. In fact, the exact value is the number the sum approaches as the number of slices goes to infinity. This is called the Riemann sum approach, and essentially, it chops the integral up in nice rectangular bits. If we wanted to approximate an integral, one way would be to chop it up into lots of rectangles, and add their areas. If we want a closer approximation, then we just make finer slices.

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