"Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions"

My paper on computation of confluent and Gauss hypergeometric functions, from a project that started in Spring 2009, finally is not only out in a journal but now even has coordinates (volume, page numbers, etc.) Here are the details.

Title: Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions

Authors: John W. Pearson, Sheehan Olver, and Mason A. Porter

Abstract: The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. Themethods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide "roadmaps" with our recommendation for which methods should be used in each situation.

Our Matlab code is available from this website.

Was the Trapezoid Rule Invented in 1994?

Well, no.

But this 1994 paper claims to have invented this method, and it has 143 citations according to google scholar.

I tried quickly via google to figure out the earliest use of the trapezoid rule, but I was unable to do so. Let me know if you happen to know how old the rule actually is. I will remark, though, that I used the rule in high school in 1993. :)

(Tip of the cap to Karen Daniels.)

Overheard in ACM 210a

In ACM 210a today, Professor Tom Hou said, "You don't want to jump the shock."

This is my quote of the day, and it will require some unraveling to convey the full significance.

First of all, it immediately made me think of the phrase jump the shark, an appreciation of which is required to fully enjoy the quote above. If you don't know what that phrase means -- and most of you should because I've discussed it on my blog before -- you owe it to yourself to take a look at the wikipedia entry on the other side of the link I provided.

The topic of the class is numerical methods for partial differential equations, and in the class of systems we're studying right now shocks are very common. When designing numerical schemes to deal with systems that can have this stuff, they typically have to be lower-order near the shock because if you're going to use nearby points for interpolation, you don't want points on both sides of the shock. That is, you don't want to jump the shock. Thus, Hou's comment (which was completely intentional) makes absolutely perfect sense scientifically. Moreover, he is as far as I can tell completely unaware of the term "jumping the shark," so it seems that he doesn't realize just how funny that comment is.

If I ever discuss the numerics of such systems in a talk, I am so going to use the picture of the Fonz jumping the shark on one of my slides. Maybe only one or two people in the audience will get the joke because it admittedly requires a reference leap, but it is bloody awesome!