Reversing recurrence relationships

  • #1
ognik
643
2
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)Pl+1 + lPl - (2l+1)xPl=0
Should I evaluate this in the 'forward' direction, by solving for Pl+1, and in the reverse direction by solving for Pl-1? I am also struggling for some intuition as to what the difference(s) may be?
 
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  • #2
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 ...
 
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