Maxwellian Probability Distribution Function
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Inverse Series Method with Padé Approximants
We can attempt to fix the problem encountered above by extending
the reach of the inverse series solutions. Often, when a truncated
power series does not perform well beyond a particular point (perhaps
because the function that the series approximates has a singularity
that limits the radius of convergence) the technique of Padé
approximants can make use of the data contained in the series coefficients
in a more robust fashion. Padé approximants are rational
function approximations to a given function. Mathematica
contains a function that creates Padé approximants in the
standard addon package Calculus`Pade`.
First load the package.
Here is the usage message for Pade:
So we define a function similar to InverseSeriesMaxwellCDF
which makes use of Padé approximants. Here we make one of
many possible choices by setting the degree of the numerator polynomial
equal to that of the denominator.
There is one subtle point that involves the fact that
gives
times a series in .
The Padé algorithm needs a series in integer powers of
the variable. While we could factor out the
and make a variable substitution prior to creating the Padé
approximant, it is easier to just use the series expression for
this special point.
Let's build the function. First make the Padé approximants
to the inverse series.
Now append the expression for the series around 0.
Here is the function.
Let's see how it compares with the InterpolationFunction
approach.
The numerical problems near zero seem to have been resolved.
Here is a plot of the difference between the Padé method
and the InterpolatingFunction
method.
Here is a timing test. The result is comparable to that for InverseSeriesMaxwellCDF
above.
