You can tell when you need to add more terms to the series when the eigenvalue line starts to curve up or down, then increase the terms and proceed.
------------Original Message------------
From: "Richard Hennessy"<rvh2007 at comcast.net>
To: "Barton Willis" <willisb at unk.edu>
Cc: dan.stanger at ieee.org, "Maxima List" <maxima at math.utexas.edu>
Date: Sun, Apr-13-2008 8:16 PM
Subject: Re: [Maxima] Can Maxima find closed form expression
I did this sort of thing already manually. I was plotting the spikes that stick out from the side of the attached plot, I just followed them and kept on increasing x and restricting further and further the range of y (Energy) and replotting. I guess that process could be converted to an algorithm.
Rich
------------Original Message------------
From: Barton Willis <willisb at unk.edu>
To: "Richard Hennessy" <rvh2007 at comcast.net>
Cc: dan.stanger at ieee.org, "Maxima List" <maxima at math.utexas.edu>
Date: Sun, Apr-13-2008 5:47 PM
Subject: Re: [Maxima] Can Maxima find closed form expression
-----maxima-bounces at math.utexas.edu wrote: -----
> This is from Quantum Mechanics so you are right about the origin of
> this problem, it is the solution for the time independent Shrodinger's
> eq. for x^4 potential, which I have posted before to this mailing list
> in a different issue.
The QM x^4 potential is solvable in terms of the Heun functions. That
buys you little (what's in a name?). Not enough is known about the
Heun functions to determine the eigenvalues of the x^4 potential in
any reasonable way, for example. And I don't think there are fast ways
to numerically evaluate the Heun functions either.
If you want to numerically find the solutions to the x^4 potential,
try summing the power series using big floats (forget about the Heun
function solution -- that just complicates matters). Use a running
error to track the subtractive cancellation. When the error is too
big, bailout and try again with a larger value for fpprec. You won't
win any races, but it might work better than you might guess. I've
done stuff like this for the hypergeometric functions. It's slow, but
it's accurate and dependable.
To learn about the running error, see "Accuracy and Stability of
Numerical Algorithms," by Nicholas J. Higham. I don't recall a
reference to the Heun function solution to the x^4 potential--there is
a standard reference to the Heun functions---maybe it is in that book.
Barton