Numerical (in)stability of recurrece relations

The previous post gave several examples of three-term recurrence relations for special functions. These relations can be computationally useful, but they have to be applied carefully. Several years ago I wrote a post on stable and unstable recurrences. In that post I show that the stability of the recurrence relation for Bessel functions produces depends on which kind of Bessel function and which direction the recurrence is applied. In the forward direction, computing higher order values from lower order values, works well for Bessel functions of the second kind Yn but not for Bessel functions of the first kind Jn. In the reverse direction, the recurrence is stable for Jn but not for Yn. I didn’t explain in that post why this is. In this post I will. Second order linear difference equations have two independent solutions, just like second order linear differential equations. For both kinds of equations, all solutions are linear combinations of the two solutions. Suppose one solution…