MHB Reversing recurrence relationships

[ognik]

A couple of times I have come across the suggestion that numerically evaluating a recursive relation in reverse can be a valuable approach. I can see this where, for example, the boundary conditions at one 'end' are inaccurate or undiscoverable. However, while the arithmetic of manipulating such equations seems simple, I wonder if I am missing something?
One example is a Legendre polynomial, given by (l+1)P_l+1 + lP_l - (2l+1)xP_l=0
Should I evaluate this in the 'forward' direction, by solving for P_l+1, and in the reverse direction by solving for P_l-1? I am also struggling for some intuition as to what the difference(s) may be?

 

[ognik]

After writing a few programs to compare, I am satisfied that when a numeric method has clear '3 points', then I can evaluate this in the 'forward' direction, by solving for Pl+1, and in the reverse direction by solving for Pl-1. In some other methods where steps are only visible as +(some step size, like h), then we can 'reverse' direction by using -h. This was all intuitively obvious, I just wanted confirmation I wasn't missing anything else ...