Maxwellian Probability Distribution Function
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Inverse Series Method
An alternative to the point sampling numerical method is to perform
series expansions of
and then invert these series using the InverseSeries
function. These series need to be computed at several points to
cover the region of interest. Here is an incomplete example: it
almost works, but fails in certain cases. We leave it as a research
project for the reader to improve upon this method so that it is
more reliable.
Here is a series expansion of MaxwellCDF
to third order about the point 1.
The InverseSeries function can be used to invert this.
The result of Series
(or InverseSeries)
is a SeriesData
object, the first part of which is the expansion parameter, the
second part the expansion point, and the third part is the list
of coefficients (Mathematica formats this in the output to
look like the standard representation of a series).
This is the expansion parameter for the inverse series
This is the expansion point.
And here is the list of coefficients in the series expansion of
the inverse function.
Using this information we can write a function that generates
the inverse series for the MaxwellCDF.
With forethought we define it to return a list that has as its first
element the expansion point and the series in the second element.
Normal is used
on the series to change it into a polynomial.
Here is an example.
This creates a table of these inverse series starting with a uniformly
spaced set of points for the MaxwellCDFSeries.
However, the resulting inverse series expansions' points are not
uniformly spaced. This is no surprise: to get a uniformly spaced
set of points we would need to have solved the inverse problem first.
Here is a timing comparison between the InverseSeries method and
the InterpolatingFunction method. The InverseSeries approach is
considerably slower.
Some of the speed loss can be recovered by compiling the function.
Let's take a look at the functions
There appear to be some numerical problems near zero.
The exercise for the interested reader is to track down the source
of the numerical instability here. Presumably it is related to the
radii of convergence of the inverse series that we have used.
