09/16/2009, 11:32 AM
The interpolation you describe is what Ansus also uses as Newton/Lagrange interpolation. (The interpolation polynomial is uniquely determined whether you use Newton or Lagrange formula or calculate it directly as solving the matrix equation like you do.)
I showed that this is interpolation (which only works for b<=e^(1/e)) is equivalent to the regular iteration at the lower (attracting) fixed point.
In so far the interpolation is "correct".
There are two problems that occur in your description with the logarithm.
1. the logarithm has no powerseries at 0 (perhaps better try to interpolate log(x+1))
2. the interpolation method seems only to converge if \( f^{[n]} \) tends to a limit. Which the logarithm does not satisfy.
I showed that this is interpolation (which only works for b<=e^(1/e)) is equivalent to the regular iteration at the lower (attracting) fixed point.
In so far the interpolation is "correct".
There are two problems that occur in your description with the logarithm.
1. the logarithm has no powerseries at 0 (perhaps better try to interpolate log(x+1))
2. the interpolation method seems only to converge if \( f^{[n]} \) tends to a limit. Which the logarithm does not satisfy.