Re: Minimal maximum eigenvalue in closed form?
- To: mathgroup at smc.vnet.net
- Subject: [mg58371] Re: [mg58315] Minimal maximum eigenvalue in closed form?
- From: Pratik Desai <pdesai1 at umbc.edu>
- Date: Tue, 28 Jun 2005 21:57:03 -0400 (EDT)
- References: <200506280913.FAA05092@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Paul Abbott wrote: >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 > > > Whoops! Forgot the most important part at the end \!\(<< LinearAlgebra`MatrixManipulation`\[IndentingNewLine] \(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 - 2 x[1], 1 - x[2], 1 - x[3]}, {1, 1 - x[5], \(-x[2]\) - x[6] + 1, 1 - x[2], 1 - 2 x[2], 1 - x[3]}, {1, 1, 1 - x[6], 1 - x[3], 1 - x[3], 1 - 2 x[3]}} /. {x[1] -> Subscript[x, 1], x[2] -> Subscript[x, 2], x[3] -> Subscript[x, 3], x[4] -> Subscript[x, 4], x[5] -> Subscript[x, 5], x[6] -> Subscript[x, 6]};\)\[IndentingNewLine] mat1 = TakeMatrix[mat, {1, 1}, {6, 6}]\[IndentingNewLine] \(expr2 = \[Lambda]\ *IdentityMatrix[6] - mat1 // Det;\)\[IndentingNewLine] NSolve[expr2 == 0, {\[Lambda], x\_1, x\_2, x\_3, x\_4, x\_5, x\_6}]\) Best regards Pratik -- Pratik Desai Graduate Student UMBC Department of Mechanical Engineering Phone: 410 455 8134
- References:
- Minimal maximum eigenvalue in closed form?
- From: Paul Abbott <paul@physics.uwa.edu.au>
- Minimal maximum eigenvalue in closed form?