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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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