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MathGroup Archive 2001

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fixed point in a function of two variables

  • To: mathgroup at smc.vnet.net
  • Subject: [mg28336] fixed point in a function of two variables
  • From: "Higinio Ramos" <higra at gugu.usal.es>
  • Date: Thu, 12 Apr 2001 02:18:01 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

I want to obtain the fixed point of a process that is supposed to be 
convergent. The Mathematica function FixedPoint also works with 
functions of two variables.Even I have added  a test function so that it 
finalizes when the difference is minor than 0.001. But I must abort the 
process because it enters a curl and there are two points that are 
repeated indefinitely as it is observed with NestList. Does anybody know 
how I can obtain the fixed point?

Thanks for your help.


r = 0.05; b0 = 0.03 10^(-3); b1 = 0.066 10^(-3); n = 500; T0 = 
80; Ta = 20;
U = 2 Pi n r/60;
e = (b1 - b0)/2;
R = r + (b0 + b1)/2;
b = Sqrt[R^2 - e^2 Sin[t]^2] + e Cos[t] - r;
sol = Solve[ 6 r  mu U NIntegrate[1/b^2, {t, 0, 2Pi}] -  r 12 mu Q 
NIntegrate[1/b^3, {t, 0, 2Pi}] == 0, Q];

Q = Q /. sol[[1]]
 \!\(\[Mu][T_] 
    10\^\(\(-3\) + 10\^\(\(-0.0565317247667186`\) + 
0.005230675060360752`\ \
\((120 - T)\)\)\)\)  

newT[Pwf2_] :=
  If[2 r <= 0.1, (-Pwf2 + 20(35 + 30 r)10^(-3) Ta)/(20(35 + 30 
r)10^(-3)),
    (-Pwf2 + 20(25 + 40 r^2 10^3)10^(-3) Ta)/(20(25 + 40 r^2 
10^3)10^(-3))]

fonRep[{T_, mu_}] :=
  (\[Tau][t_] := -4 mu U /b + 6 mu Q/b^2;
M = r^2NIntegrate[\[Tau][t], {t, 0, 2Pi}];
    Pwf = M U/r; {newT[Pwf], \[Mu][newT[Pwf]]})


test1[x_, y_] := Abs[x[[1]] - y[[1]]] < .0001

{Tfinal, \[Mu]final} = FixedPoint[fonRep, {Ta, \[Mu][Ta]}, SameTest -> 
test1]

In[105]:=
NestList[fonRep, {Ta, \[Mu][Ta]}, 10]
Out[105]=
{{20, 0.846763}, {67213.4,
    
0.00100000000000000000000000000000000000000000000000000000000000000000000
0\
0000000000000000000000000000000000000000000000000000000000000000000000000
00000\
0000000000000000000000000000000000000000000000000000000000000000000000000
00000\
0000000000000000000000000000000000000000000000000000000000000000000000000
00000\
000000000000000000000000000000000000000000000002300310171744}, {99.3532, 

    0.0133601}, {1080.17, 0.00100002}, {99.3548, 0.0133594}, {1080.12,
    0.00100002}, {99.3548, 0.0133594}, {1080.12, 0.00100002}, {99.3548,
    0.0133594}, {1080.12, 0.00100002}, {99.3548, 0.0133594}}



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