The previous post gave respective examples of three-term recurrence relations for typical functions. These relations tin beryllium computationally useful, but they person to beryllium applied carefully.
Several years agone I wrote a station connected stable and unstable recurrences. In that station I show that the stableness of the recurrence narration for Bessel functions produces depends connected which benignant of Bessel usability and which guidance the recurrence is applied.
In the guardant direction, computing higher bid values from little bid values, useful good for Bessel functions of the 2nd benignant Yn but not for Bessel functions of the first benignant Jn. In the reverse direction, the recurrence is unchangeable for Jn but not for Yn.
I didn’t explicate successful that station why this is. In this station I will.
Second bid linear quality equations person 2 independent solutions, conscionable for illustration 2nd bid linear differential equations. For some kinds of equations, each solutions are linear combinations of the 2 solutions. Suppose 1 solution grows pinch n and the different decays. You whitethorn want to compute the decaying solution, but successful doing truthful you mightiness prime up a mini constituent of the increasing solution owed to rounding error. This post illustrates this phenomena for differential equations, and this post illustrates it for quality equations.
When you look astatine a crippled of Bessel functions successful a matter book, you’ll astir apt spot a fewer land of Jn(x) andYn(x) for a fewer mini values of n. The functions look to behave astir the aforesaid way, for illustration sine and cosine. And that’s true, as functions of x.
But it’s not existent for Jn(x) andYn(x) arsenic functions of n for fixed x. As n increases, Jn(x) decays to zero and Yn(x) goes disconnected to −∞.
That’s the root of numerical instability. And location will beryllium akin instability problems for different recurrences wherever the ratios of the 2 independent solutions goes to zero aliases infinity arsenic a usability of n.
There are techniques for computing the solution that does not diverse, the alleged minimal solution, specified arsenic Miller’s algorithm mentioned here.
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