Dear maxima users,
I've been working on some calculations that eventually lead to a 5x5
matrix, whose components can be complex. The next step is to calculate
the eigenvalues and possibly the eigenvectors.
The built-in maxima functions: eigenvalues and eigenvectors
immediately fail, even when the matrix has all real components. It
says "solve is unable to find the roots of the characteristic
polynomial".
I looked at "dgeev" which uses lapack, and it worked superbly when the
matrix components are real, but gave me errors when they were complex,
as expected. Sadly dgeev only works for integers and floating point.
Is there any other tool/function that will compute these things
numerically? I deally I would want to later plot each eigenvalue as a
function of one or two parameters.
Perhaps maxima is not the appropriate tool to do this anymore (my
original hope was to compute things analytically, but that's pretty
much out the window now).. care to suggest a good way to go?
As an additional thing, in an earlier stage of my work (which proved
wrong), I was able to reduce the characteristic polynomial to a 4th
order equation. Wikipedia says, analytical solutions are possible
(http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method)
through something known as the Ferrari's method. However, when I tried
using 'solve' with this quartic polynomial, maxima just couldn't do
it. After many minutes, maxima simply crashed...(it was disconnected
from WxMaxima)
I have to say that the coefficients of the polynomial were quite
lengthy, but I feel it should have worked...Naturally when I try a
general quartic polynomial like listed in the wikipedia article,
things work fine, though I get warned the expression is too long to
display. What are the limitations here?...it really is just a lot of
algebra...
Now, in the case of quintic polynomials, which brings me back to my
original question, 'solve' can't even do this:
solve( z^5 -z -1 = 0 , z);
This is a well known quintic polynomial whose roots cannot be obtained
analytically by any means, but they should be possible, numerically.
Is there a way to get the roots using solve or other tool in maxima?
if so, can I "instruct" eigenvalues / eigenvectors to use this instead
of plain solve?
I'm using Maxima 5.21.1 on Debian Squeeze. GCL 2.6.7
Thanks for your input
Andres