Re: How to find more solutions for an periodical equation with infinity solutions
- To: mathgroup at smc.vnet.net
- Subject: [mg122928] Re: How to find more solutions for an periodical equation with infinity solutions
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Wed, 16 Nov 2011 04:47:09 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <j9tg76$n8d$1@smc.vnet.net>
Using Reduce will tell you much more about possible solutions (probably everything): Reduce[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]] . {x, y, z}, \[Theta]] (C[1] \[Element] Integers && (\[Theta] == 2*Pi*C[1] || (x == 0 && y == 0 && \[Theta] == Pi + 2*Pi*C[1]))) || ((-Pi + \[Theta])/(2*Pi) \[NotElement] Integers && x == 0 && y == 0) Your equation to be solved for \[Theta] obviously is looking for Eigenvalues == 1 of the Matrix with Alpha = 0 and Beta = 0. This can be accomplished more directly in the follwing way ev = Eigenvalues[Matrix[{0, 0, 1}, \[Theta]]] {1, Cos[\[Theta]] - I*Sin[\[Theta]], Cos[\[Theta]] + I*Sin[\[Theta]]} Reduce[ev[[1]] == 1, \[Theta]] True Reduce[ev[[2]] == 1, \[Theta]] C[1] \[Element] Integers && \[Theta] == 2*Pi*C[1] Reduce[ev[[3]] == 1, \[Theta]] C[1] \[Element] Integers && \[Theta] == 2*Pi*C[1] PS: I don't see the 5 or 6 solutions you mentioned, but only 3. Regards, Wolfgang "Gy Peng" <hitphyopt at gmail.com> schrieb im Newsbeitrag news:j9tg76$n8d$1 at smc.vnet.net... > Dear all, > > I have a matrix defined as: > Matrix[{\[Alpha]_, \[Beta]_, \[Gamma]_}, \[Theta]_] := {{\[Alpha]^2 \ > (1 - Cos[\[Theta]]) + > Cos[\[Theta]], \[Alpha] \[Beta] (1 - > Cos[\[Theta]]) - \[Gamma] Sin[\[Theta]], \[A lpha] \[Gamma] > (1 - > Cos[\[Theta]]) + \[Beta] Sin[\[Theta]]}, {\[Alpha] \[Beta] (1 > \ > - Cos[\[Theta]]) + \[Gamma] Sin[\[Theta]], \[Beta]^2 (1 - > Cos[\[Theta]]) + > Cos[\[Theta]], \[Beta] \[Gamma] (1 - > Cos[\[Theta]]) - \[Alpha] Sin[\[Theta]]}, {\[Alpha] \[Gamma] \ > (1 - Cos[\[Theta]]) - \[Beta] Sin[\[Theta]], \[Beta] \[Gamma] (1 - > Cos[\[Theta]]) + \[Alpha] Sin[\[Theta]], \[Gamma]^2 (1 - > Cos[\[Theta]]) + Cos[\[Theta]]}} > > Solve[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]].{x, y, z}, \[Theta]] > > I know this equation is periodical and has infinity solutions. So, > Mathematica only gave me only one solution: {{\[Theta] -> 0}} and > show me the message: > > Solve::ifun: Inverse functions are being used by Solve, so some > solutions > may not be found; use Reduce for complete solution information. >> > > My question is, how could I see for example first 6 solution, because > I > know the first 5 or 6 solutions should be different and then > repeat themselves periodically. What should I do to find the first 5 > or 6 > solutions? > > Thank you very much!