Saturday, July 14, 2012

Maxima/wxMaxima - solving for the roots of a transcendental equation

Ok, so something is wrong with my Python.  I think Spyder is giving me some fits in Ubunt for somereason.  Anyways, luckily I have Maxima and Sage to fall back on.  In order to solve for the roots of a transcendental equation, I provide a following example.

The equation I want to solve is:

\[ \tan \left( \tfrac{1}{2} C d^2 \right) - \tan \left( \tfrac{1}{2} C a^2 \right) = 0 \]

where \( \deta \) is given.  I first plotted it to make sure it did in fact show trends of a root or roots for \( \delta = 0.5\).

eqn(x):= tan(0.5*x*(delta^2)) - tan(0.5*x);
plot2d(eqn, [x, 0, 20],[y, -1, 1]);

So, yup, it definitely has roots.  The first root is actually at zero, but this is trivial in my case.  So I am interesting in the second root.  It seems to be around 8.  I choose the interval for the root finder to look on as 7.8 to 8.5.

find_root(tan(0.5*x*(delta^2)) - tan(0.5*x), x, 7.8, 8.5);

So the root found is the bold number above!

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