[EM] Interpolation revisions, detail clarification
email9648742 at gmail.com
Tue Jul 3 04:25:59 PDT 2012
I suggested using, as an interpolating function, A*e^(-kx). But, more
constants would pass the function through more data points, for a more
accurate interpolation. So I'd suggest adding a constant:
A*e^(-kx)+B. To pass the function through 3 data points.
Warren was using one exponential function for the entire range, instead of
using a different one to interpolate, with a few points, for each interval
between two consecutive integers. He said that that exponential wouldn't
really be an exact measure of the frequencies of states of various sizes.
And, in fact, I don't know if exponentials could give a good fit at all,
for intervals in some parts of the overall range of interest.
So maybe a polynomial interpolation would be needed for some regions. If
so, it might even be desirable everywhere. As before, it would be a
different interpolating polynomial for each interval between consecutive
When I said to sum P(s) between S(a) and S(b), I didn't say anything about
how that would be done.
P(s) is the inverse of the interpolating function. As I said, the inverse
of an exponential is a logarithmic function. The inverse of some polynomial
functions can't be written. I believe that's true of every polynomial of
degree higher than two.
One could use a quadratic, of course.
But, evaluating a definite integral of a function whose inverse is easily
antidifferentiated, isn't a problem. So, as long as the interpolating
function itself is easily antidifferentiated, there won't be a problem.
Aside from that, the logarithmic functions resulting from the exponential
interpolating functions that I suggested can be antidifferntiated by a
substitution and an integration by parts.
When the Weighted-Webster procedure that I described is applied to the S(x)
interpolating function, kx, which would be consistent with a flat
probability distribution, the roundoff point, R, is a + 1/2. In other
words, Webster's method.
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