Numerical (In)Stability of Recurrence Relations
This article explains the mathematical phenomenon of numerical instability in three-term recurrence relations, specifically regarding Bessel functions. It details how rounding errors can propagate when computing solutions that decay versus those that grow.
Why it matters
Understanding numerical stability is critical for scientists and engineers developing software that relies on high-precision mathematical computations.
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 Y n but not for Bessel functions of the first kind J n . In the reverse direction, the recurrence is stable for J n but not for Y n .
I didn’t explain in that post why this is. In this post I will.
Get smarter about the news
Sign up free for a feed built around what you actually care about, Dive Deeper research on any story, and the full text of every article.
Create free accountAlready have an account? Sign in