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Topic: limits on symbol eigenvalues?
Replies: 2   Last Post: May 12, 2014 12:44 AM

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David Bailey

Posts: 714
Registered: 11/7/08
Re: limits on symbol eigenvalues?
Posted: May 12, 2014 12:44 AM
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On 04/06/2004 10:14, Uwe Brauer wrote:
> Hello
>
> I just started using mathematica. When I tried to calculate the
> symbolic eigenvalues of a 16x16 matrix mathematica told me it couldn't
>
> Is there a restriction?
>
> Thanks
>
> Uwe Brauer
>

Not every symbolic problem that you can pose has a symbolic solution.
For example, some symbolic integrals don't have symbolic solutions -
likewise for differential equations.

A symbolic eigenvalue problem of order N involves solving an N'th order
polynomial equation. Specific cases can be solved, but the general case
cannot be solved for N>=5. This restriction can in theory be relaxed (I
am not sure by how much) by the use of theta functions, though the
symbolic answers are impossibly large.

Even when a symbolic solution is possible, it may not be desirable
because it is excessively complicated, and possibly numerically unstable
if the coefficients are subsequently replaced by numbers. To see what I
mean, try evaluating:

Solve[a x^4 + b x^3 + c x + d == 0, x]

David Bailey
http://www.dbaileyconsultancy.co.uk





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