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MathGroup Archive 2005

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Minimal maximum eigenvalue in closed form?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58315] Minimal maximum eigenvalue in closed form?
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 28 Jun 2005 05:13:14 -0400 (EDT)
  • Organization: The University of Western Australia
  • Sender: owner-wri-mathgroup at wolfram.com

Here is an interesting exercise: compute the minimal maximum eigenvalue 
of the matrix (arising in a semidefinite programming problem)

 mat = 
 {
  {1, 1 - x[4], 1 - x[4], 1 - x[4], 1, 1},
  {1 - x[4], 1, 1 - x[5], -x[1] - x[5] + 1, 1 - x[5], 1},
  {1 - x[4], 1 - x[5], 1, 1 - x[1] - x[6], 1 - x[2] - x[6], 1 - x[6]},
  {1 - x[4], 1-x[1] -x[5], 1-x[1] -x[6], 1 - 2x[1], 1 - x[2], 1 - x[3]},
  {1, 1 - x[5], -x[2] - x[6] + 1, 1 - x[2], 1 - 2x[2], 1 - x[3]}, 
  {1, 1, 1 - x[6], 1 - x[3], 1 - x[3], 1 - 2x[3]}
 };

in closed form. This is reminiscent of the sort of problems given in the 
SIAM 100 digit challenge, see
 
  mathworld.wolfram.com/Hundred-DollarHundred-DigitChallengeProblems.html

Numerically, the answer is 1.5623947722331... 

It can be shown that the exact answer can be expressed as the root of a 
6th order polynomial. Does anyone have an elegant way of obtaining the 
solution (and also the values of x[1] through x[6])?

Cheers,
Paul

-- 
Paul Abbott                                      Phone: +61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul
        http://InternationalMathematicaSymposium.org/IMS2005/


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