Re: problem with Quaternion polynomial root solver
- To: mathgroup at smc.vnet.net
- Subject: [mg68352] Re: problem with Quaternion polynomial root solver
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Wed, 2 Aug 2006 05:24:08 -0400 (EDT)
- References: <eahtq5$p0b$1@smc.vnet.net> <eakd82$qhf$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In an attempt to find a Quaternion Pisot like solution I attempt to find fixed point solutions to a 4d Quaternion Migdal-Kadanov under the impression that the 4d lattice involved would be Ising like and have a Pisot related solution as happens in two dimensions. ( well known 2d fixed point solution is b1=Log[1+Sqrt[2]]/2 which is related to the Pisot polynomial x^2-2*x-1) I get a solution in Mathematica of the form: b0=F[b1,b2,b3] but not a true fixed point numerical solution. Can anyone suggest a better way to find the fixed point solution in Mathematica. I also tried an SU(3) Migdal -Kadanov and got a recursion solution, but Mathematica refused to solve for the fixed point. \!\(Clear[t, x, y, z, ta, xa, ya, za, b0, b1, b2, b3, b0a, b1a, b2a, b3a]\n (*\ since\ the\ fixed\ point\ of\ the\ Migdal\ - Kadanov\ recursion\ is\ connected\ to\ a\ quadratic\ Pisot\ with\ root\ 1 + Sqrt[2]*) \n (*I' m\ trying\ \ to\ get\ a\ fixed\ point\ for\ a\ Quaternion\ Migdal - Kadanov\ to\ see\ what\ \ kind\ of\ root\ it\ involves*) \[IndentingNewLine] << Algebra`Quaternions`\n (*\ four\ variable\ spin/ magnetic\ exponential\ definition\ of\ the\ Migdal\ - Kadanov\ \ form*) \n t = Exp[b0 + b1]; ta = Exp[b0a + b1a];\n x = Exp[b0 - b1 + b2]; xa = Exp[b0a - b1a + b2a];\n y = Exp[b0 - b1 - b2 + b3]; ya = Exp[b0a - b1a - b2a + b3a];\n z = Exp[b0 - b1 - b2 - b3]; za = Exp[b0a - b1a - b2a - b3a];\n \(q = Quaternion[t, x, y, z];\)\n \(qa = Quaternion[ta, xa, ya, za];\)\n \(q2 = ExpandAll[q ** q];\)\n FullSimplify[q2 - qa]\n (*Solve\ for\ Migdal_Kadanov\ recursion*) \n Solve[{\(-\[ExponentialE]\^\(b0a + b1a\)\) - \[ExponentialE]\^\(2\ b0 - 2\ \ b1 - 2\ b2 - 2\ b3\)\ \((1 + \[ExponentialE]\^\(4\ b3\) - \ \[ExponentialE]\^\(2\ \((2\ b1 + b2 + b3)\)\) + \[ExponentialE]\^\(4\ b2 + 2\ \ b3\))\) == 0, 2\ \[ExponentialE]\^\(2\ b0 + b2\) - \[ExponentialE]\^\(b0a - \ b1a + b2a\) == 0, 2\ \[ExponentialE]\^\(2\ b0 - b2 + b3\) - \[ExponentialE]\^\ \(b0a - b1a - b2a + b3a\) == 0, 2\ \[ExponentialE]\^\(2\ b0 - b2 - b3\) - \[ExponentialE]\^\(b0a - b1a - b2a - b3a\) == 0}, {b0a, b1a, b2a, b3a}]\n FullSimplify[{b0 - \((1\/8\ \((2\ Log[2\ \[ExponentialE]\^\(2\ b0 + b2\)] + \ Log[2\ \[ExponentialE]\^\(2\ b0 - b2 - b3\)] + Log[2\ \[ExponentialE]\^\(2\ b0 - b2 + b3\)] + 4\ Log[\(-\[ExponentialE]\^\(2\ b0 - 2\ b1 - 2\ \ b2 - 2\ b3\)\)\ \((1 + \[ExponentialE]\^\(4\ b3\) - \[ExponentialE]\^\(4\ b1 \ + 2\ b2 + 2\ b3\) + \[ExponentialE]\^\(4\ b2 + 2\ b3\))\)])\))\), b1 - \ \((1\/8\ \((\(-2\)\ Log[2\ \[ExponentialE]\^\(2\ b0 + b2\)] - Log[2\ \[ExponentialE]\^\(2\ b0 - b2 - b3\)] - Log[2\ \ \[ExponentialE]\^\(2\ b0 - b2 + b3\)] + 4\ Log[\(-\[ExponentialE]\^\(2\ b0 - \ 2\ b1 - 2\ b2 - 2\ b3\)\)\ \((1 + \[ExponentialE]\^\(4\ b3\) - \ \[ExponentialE]\^\(4\ b1 + 2\ b2 + 2\ b3\) + \[ExponentialE]\^\(4\ b2 + 2\ b3\))\)])\))\), b2 - \((1\/4\ \((2\ Log[2\ \ \[ExponentialE]\^\(2\ b0 + b2\)] - Log[2\ \[ExponentialE]\^\(2\ b0 - b2 - b3\)] - Log[2\ \[ExponentialE]\^\(2\ b0 - b2 + b3\)])\))\), b3 - \ \((1\/2\ \((\(-Log[2\ \[ExponentialE]\^\(2\ b0 - b2 - b3\)]\) + Log[2\ \ \[ExponentialE]\^\(2\ b0 - b2 + b3\)])\))\)}]\n NSolve[{1\/8\ \((8\ b0 - 4\ Log[2] - 2\ Log[\[ExponentialE]\^\(2\ b0 + \ b2\)] - Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] - Log[\[ExponentialE]\^\(2\ b0 - b2 + b3\)] - 4\ Log[\(-\[ExponentialE]\^\(2\ b0 - 2\ b1 - 2\ \ b2 - 2\ b3\)\)\ \((1 + \[ExponentialE]\^\(4\ b3\) - \[ExponentialE]\^\(2\ \(( 2\ b1 + b2 + b3)\)\) + \[ExponentialE]\^\(4\ b2 + 2\ b3\))\)])\) == 0, 1\/8\ \((8\ b1 + Log[16] + 2\ Log[\[ExponentialE]\^\(2\ b0 + b2\)] + Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] + Log[\[ExponentialE]\ \^\(2\ b0 - b2 + b3\)] - 4\ Log[\(-\[ExponentialE]\^\(2\ b0 - 2\ b1 - 2\ b2 - 2\ b3\)\)\ \ \((1 + \[ExponentialE]\^\(4\ b3\) - \[ExponentialE]\^\(2\ \((2\ b1 + b2 + b3)\ \)\) + \[ExponentialE]\^\(4\ b2 + 2\ b3\))\)])\) == 0, 1\/4\ \((4\ b2 - 2\ \ Log[\[ExponentialE]\^\(2\ b0 + b2\)] + Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] + Log[\[ExponentialE]\^\(2\ b0 - b2 + b3\)])\) == 0, b3 + \ 1\/2\ \((Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] - Log[\[ExponentialE]\^\(2\ \ b0 - b2 + b3\)])\) == 0}, {b0, b1, b2, b3}]\) Try for further solution: \!\(b0 = Log[\(1.`\ \[ExponentialE]\^\(b1 + b2 + b3\)\)\/\@\(\(-2.`\) - 2.`\ \ \[ExponentialE]\^\(4\ b3\) + 2.`\ \[ExponentialE]\^\(4\ b1 + 2\ b2 + 2\ b3\) \ - 2.`\ \[ExponentialE]\^\(4\ b2 + 2\ b3\)\)]\n NSolve[{1\/8\ \((8\ b1 + Log[16] + 2\ Log[\[ExponentialE]\^\(2\ b0 + b2\)] \ + Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] + Log[\[ExponentialE]\^\(2\ b0 - \ b2 + b3\)] - 4\ Log[\(-\[ExponentialE]\^\(2\ b0 - 2\ b1 - 2\ b2 - 2\ b3\)\)\ \ \((1 + \[ExponentialE]\^\(4\ b3\) - \[ExponentialE]\^\(2\ \((2\ b1 + b2 + b3)\ \)\) + \[ExponentialE]\^\(4\ b2 + 2\ b3\))\)])\) == 0, 1\/4\ \((4\ b2 - 2\ Log[\[ExponentialE]\^\(2\ b0 + b2\)] + \ Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] + Log[\[ExponentialE]\^\(2\ b0 - b2 + \ b3\)])\) == 0, b3 + 1\/2\ \(( Log[\[ExponentialE]\^\(2\ b0 - b2 - b3\)] - Log[\[ExponentialE]\ \^\(2\ b0 - b2 + b3\)])\) == 0}, {b1, b2, b3}]\)