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/

**Follow-Ups**:**Re: Minimal maximum eigenvalue in closed form?***From:*Pratik Desai <pdesai1@umbc.edu>

**Re: Minimal maximum eigenvalue in closed form?***From:*Pratik Desai <pdesai1@umbc.edu>