Numerical footnote

Yesterday’s post said that that you could construct a chain of linear relationships between the hypergeometric function

F(a, b; c; z)


F(a+i, b+j; c+k; z)

for integers i, j, and k. Toward the end of the post I said that this could be used to speed up programs by computing function values from previous values rather than from scratch. This is true, but you need to check whether there are numerical issues.

You have to be careful when using recurrence relations numerically. It’s common for a recurrence to be numerically stable in one direction but unstable in the opposite direction. I wrote about this here and gave examples.

The stability problems for recurrence relations are the discrete analog of the more familiar problem of sensitive dependence on initial conditions for differential equations described here.