Counting Trees and Solving Equations

My latest three posts were motivated by a numerical analysis problem, but I believe they would be interested to a wider audience. They’re not about numerical analysis per se; they’re about a solving a set of equations that come out of numerical analysis.

The title of this article has a double meaning. “Solving equations” refers to solving a set of algebraic equations that are necessary in order to develop numerical methods of solving differential equations.

The fourth order Runge-Kutta method is a standard textbook method for numerically solving differential equations. You never see it derived in an undergraduate textbook, and for good reason: the derivation is complicated.

One step in the design of fourth order RK methods is solving a system of 8 equations in 10 unknowns. I was curious whether Mathematica would solve the system, and what it would take to add constrains to make the solution unique. See more here.

The next two posts generalize on this.

Fourth order RK methods are also four stage RK methods. But in general the number of stages must be greater than the order. What is exactly is a stage and what exactly is the order? See here.

How many equations and how many variables would there be in designing higher order RK methods? The answer involves a surprising connection.

The number of equations required in designing an RK method with s stages is closely related to the number of distinct rooted trees with s nodes. So there’s a direct link between the pure math problem of counting rooted trees with the applied math problem of solving differential equations. More here.

Enjoy reading. Have a good weekend.