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Re: limits on symbol eigenvalues?

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  • Subject: [mg132717] Re: limits on symbol eigenvalues?
  • From: David Bailey <dave at removedbailey.co.uk>
  • Date: Mon, 12 May 2014 00:44:03 -0400 (EDT)
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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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