Re: R: Problem with circles in complex plane

*To*: mathgroup at smc.vnet.net*Subject*: [mg61133] Re: R: [mg61095] Problem with circles in complex plane*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 11 Oct 2005 03:20:34 -0400 (EDT)*References*: <DFBFB541-1F25-4B4C-8967-7E467DBB59B4@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

I notice that there is an even simpler way: Solve[(ComplexExpand[(Re[CF] - Re[Cg])^2 + (Im[CF] - Im[Cg])^2, {P, G}] - (RF + Rg)^2 /. rul) == 0, g] Potential solution -0.377592629328396048282835483011872267671725623879`32.97373448684384 (possibly discarded by verifier) should be checked by hand. May require use of limits. {{g -> 0.9136201753885568}} This makes fewer assumptions than my earlier message but produces a curious message that I have never seen before. On 10 Oct 2005, at 22:09, Andrzej Kozlowski wrote: > > On 10 Oct 2005, at 19:05, Daniele Lupo wrote: > > > >>> >>> >>> >> >> The fact is that I must to find a symbolic solution: for every P, >> Q, G that >> satisfy myu conditions (absolute value, real number and so on), I >> want to >> find the g value that define the tangent circle. >> >> I hope that's more clear... >> >> > > What is not clear is how you managed to get a symbolic formula, if > you did. I have not tried it very hard, but on quick glance I do > not think there is any way to do so "reliably". Solve will only > give you reliable general solutions for algebraic equations: > basically polynomial and rational functions. It can also solve > equations with radicals but this is much harder and time consuming > and may produce extraneous solutions. Now, the equation that you > get if you apply Solve without any massaging is transcendental (it > involves the function Abs) and when Mathematica solves such > equations it can fail to find all solutions. This is normal. So if > you want to get all solutions you need to convert the equation to a > basically algebraic form. This can be done, if you make some > assumptions. I will assume that we know that there exists an real > solution between 0 and 1. Once we do that we can proceed as follows. > > We start with your definitions: > > In[1]:= > CF = G/(Q + 1); > > In[2]:= > RF = Sqrt[Q^2 + (1 - Abs[G]^2)*Q]/(Q + 1); > > In[3]:= > Cg = (g*Conjugate[P])/(1 - Abs[P]^2*(1 - g)); > > In[4]:= > Rg = (Sqrt[1 - g]*(1 - Abs[P]^2))/(1 - Abs[P]^2*(1 - g)); > > In[5]:= > rul = {G -> -0.4608904699810983 + 0.11491290040984217*I, Q -> 0.3, > P -> -0.8363463602974097 + 0.16256926406081632*I} > > Out[5]= > {G -> -0.4608904699810983 + 0.11491290040984217*I, Q -> 0.3, > P -> -0.8363463602974097 + 0.16256926406081632*I} > > > Now, I am assuming that g is a solution of the equation > > Abs[CF - Cg]^2 - (RF + Rg)^2==0 (I am trying to minimize the number > of square roots) > > and moreover than it is a real solution and that it lies between 0 > and 1. This makes it possible to transform the equation as follows: > > > eq = Simplify[ComplexExpand[Abs[CF - Cg]^2 - (RF + Rg)^2, > {P, G}, TargetFunctions -> {Re, Im}], 0 < g < 1] /. > rul > > > ((0.16256926406081632*g)/(0.7259040000000001*(g - 1) + > 1) + 0.08839453877680166)^2 + > ((0.8363463602974097*g)/(0.7259040000000001*(g - 1) + > 1) - 0.35453113075469095)^2 + > (0.07512861721599995*(g - 1))/ > (0.7259040000000001*(g - 1) + 1)^2 - > (0.2394020778504835*((g - 1)^2)^(1/4))/ > (0.7259040000000001*(g - 1) + 1) - 0.19071745562130174 > > > As you can see this is now an algebraic equation. We can now use > NSolve: > > In[7]:= > NSolve[eq == 0, g] > > Out[7]= > {{g -> 0.9136201753885588}} > > > >> but, as I said, but I think that a symbolic solution >> must exist. In this case, I don't understand why with FindRoot I >> find the >> solution (considering numeric approximation), while with Solve, >> that's >> symbolic method, I obtain wrong values, because Solve gives me >> solutions. >> Maybe I think that Solve gives a wrong result in this case? A >> Solve bug? I >> don't want to think this. >> >> > > Solve does not find all solutions because it can't reliably solve > transcendental equations. It gives you some solutions, which may be > actually extraneous. > > > >> equations cannot be solved, I use FindRoot but, if Solve gives me >> solutions, >> I don't understand why these don't match my problem. If solution is >> symbolic, why I don't obtain the right solution? It's not a numeric >> problem... >> >> >> > > You are right, it is not a problem with numerical stability, it is > just a problem with transcendental equations. You have ot be able > to produce an algebraic equation first before you can use Solve > reliably. > > Andrzej Kozlowski > > > > > >

**Re: Problem with circles in complex plane**

**Re: Wick like theorem and "symbolic" compilation**

**Re: R: Problem with circles in complex plane**

**Using MathLink to create a GUI**