By:Yanliu
In[]:=
pm=Compile[{{grid,_Real,2}},Module[{plate=grid},Do[plate[[a,b]]=Piecewise[{{plate[[a,b]],(a<11∧b11)∨(a11∧b≥11)},{0.,b≥11∧a≤10}},Mean[{plate[[If[a1,a,a-1],b]],plate[[If[a20,a-1,a+1],b]],plate[[a,If[b1,b,b-1]]],plate[[a,If[b20,b-1,b+1]]]}]],{a,2,19},{b,2,19}];plate]];​​
In[]:=
ListPlot3D[Last[NestList[pm,{{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,10.,9.,8.,7.,6.,5.,4.,3.,2.,0.},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}},190]]]
Out[]=
In[]:=
MatrixPlot[Last[NestList[pm,{{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,10.,9.,8.,7.,6.,5.,4.,3.,2.,0.},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}},190]]]
Out[]=
The above graphes are the results of 190 times of recurrsion, now we' ll see if this result converges. We compare the result of 189 th recurrsion and 190 th recurrsion.
In[]:=
Last[NestList[pm,{{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,10.,9.,8.,7.,6.,5.,4.,3.,2.,0.},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}},190]]-Last[NestList[pm,{{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,10.,9.,8.,7.,6.,5.,4.,3.,2.,0.},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}},189]]
Out[]=
{{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,4.54313×
-7
10
,8.42125×
-7
10
,1.13028×
-6
10
,1.29638×
-6
10
,1.33036×
-6
10
,1.23499×
-6
10
,1.02524×
-6
10
,7.26611×
-7
10
,3.72363×
-7
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,8.79785×
-7
10
,1.63101×
-6
10
,2.18958×
-6
10
,2.5121×
-6
10
,2.57888×
-6
10
,2.39496×
-6
10
,1.98899×
-6
10
,1.41012×
-6
10
,7.22812×
-7
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,1.27301×
-6
10
,2.36058×
-6
10
,3.17027×
-6
10
,3.63922×
-6
10
,3.73845×
-6
10
,3.4744×
-6
10
,2.88763×
-6
10
,2.04859×
-6
10
,1.05058×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,1.63067×
-6
10
,3.02492×
-6
10
,4.06504×
-6
10
,4.67038×
-6
10
,4.80287×
-6
10
,4.46911×
-6
10
,3.71909×
-6
10
,2.64154×
-6
10
,1.35581×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,1.94926×
-6
10
,3.6179×
-6
10
,4.8664×
-6
10
,5.59832×
-6
10
,5.76661×
-6
10
,5.3763×
-6
10
,4.48343×
-6
10
,3.19084×
-6
10
,1.64025×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,2.22484×
-6
10
,4.13257×
-6
10
,5.56596×
-6
10
,6.41511×
-6
10
,6.62423×
-6
10
,6.19453×
-6
10
,5.18361×
-6
10
,3.70216×
-6
10
,1.90864×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,2.45275×
-6
10
,4.56064×
-6
10
,6.15353×
-6
10
,7.11103×
-6
10
,7.36924×
-6
10
,6.92327×
-6
10
,5.8264×
-6
10
,4.1878×
-6
10
,2.1718×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,2.62731×
-6
10
,4.89179×
-6
10
,6.61598×
-6
10
,7.67286×
-6
10
,7.99206×
-6
10
,7.56116×
-6
10
,6.423×
-6
10
,4.67164×
-6
10
,2.45475×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,2.74175×
-6
10
,5.11338×
-6
10
,6.93627×
-6
10
,8.08173×
-6
10
,8.47633×
-6
10
,8.10148×
-6
10
,6.98676×
-6
10
,5.19696×
-6
10
,2.81931×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,2.78848×
-6
10
,5.21061×
-6
10
,7.09311×
-6
10
,8.3112×
-6
10
,8.79375×
-6
10
,8.52216×
-6
10
,7.52077×
-6
10
,5.83069×
-6
10
,3.43511×
-6
10
,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,2.75986×
-6
10
,5.16802×
-6
10
,7.06235×
-6
10
,8.32735×
-6
10
,8.89976×
-6
10
,8.76974×
-6
10
,7.97807×
-6
10
,6.61294×
-6
10
,4.82306×
-6
10
,2.93×
-6
10
,2.05761×
-6
10
,1.54938×
-6
10
,1.19582×
-6
10
,9.20465×
-7
10
,6.91034×
-7
10
,4.91753×
-7
10
,3.1372×
-7
10
,1.51193×
-7
10
,0.},{0.,2.64969×
-6
10
,4.97226×
-6
10
,6.82134×
-6
10
,8.09419×
-6
10
,8.73779×
-6
10
,8.75161×
-6
10
,8.18833×
-6
10
,7.15586×
-6
10
,5.82857×
-6
10
,4.47699×
-6
10
,3.47253×
-6
10
,2.72653×
-6
10
,2.14348×
-6
10
,1.66433×
-6
10
,1.255×
-6
10
,8.95183×
-7
10
,5.71814×
-7
10
,2.75752×
-7
10
,0.},{0.,2.45498×
-6
10
,4.61604×
-6
10
,6.35564×
-6
10
,7.58548×
-6
10
,8.26245×
-6
10
,8.3915×
-6
10
,8.02609×
-6
10
,7.2675×
-6
10
,6.26287×
-6
10
,5.19612×
-6
10
,4.24248×
-6
10
,3.43148×
-6
10
,2.74278×
-6
10
,2.1495×
-6
10
,1.6294×
-6
10
,1.16575×
-6
10
,7.45914×
-7
10
,3.60023×
-7
10
,0.},{0.,2.1776×
-6
10
,4.10166×
-6
10
,5.66518×
-6
10
,6.79494×
-6
10
,7.4565×
-6
10
,7.65641×
-6
10
,7.44154×
-6
10
,6.89557×
-6
10
,6.13118×
-6
10
,5.27436×
-6
10
,4.43358×
-6
10
,3.65905×
-6
10
,2.96304×
-6
10
,2.34119×
-6
10
,1.78371×
-6
10
,1.28009×
-6
10
,8.20563×
-7
10
,3.96429×
-7
10
,0.},{0.,1.82524×
-6
10
,3.44291×
-6
10
,4.76753×
-6
10
,5.74098×
-6
10
,6.33645×
-6
10
,6.55982×
-6
10
,6.44825×
-6
10
,6.06573×
-6
10
,5.49474×
-6
10
,4.8232×
-6
10
,4.12659×
-6
10
,3.45223×
-6
10
,2.8227×
-6
10
,2.24499×
-6
10
,1.71783×
-6
10
,1.23623×
-6
10
,7.93776×
-7
10
,3.83833×
-7
10
,0.},{0.,1.41136×
-6
10
,2.66516×
-6
10
,3.6978×
-6
10
,4.46612×
-6
10
,4.95035×
-6
10
,5.15483×
-6
10
,5.10646×
-6
10
,4.85068×
-6
10
,4.44487×
-6
10
,3.94921×
-6
10
,3.41639×
-6
10
,2.88402×
-6
10
,2.3743×
-6
10
,1.89765×
-6
10
,1.45698×
-6
10
,1.05086×
-6
10
,6.75695×
-7
10
,3.26982×
-7
10
,0.},{0.,9.54367×
-7
10
,1.80361×
-6
10
,2.50586×
-6
10
,3.03276×
-6
10
,3.3713×
-6
10
,3.52417×
-6
10
,3.50855×
-6
10
,3.35324×
-6
10
,3.09426×
-6
10
,2.7693×
-6
10
,2.41199×
-6
10
,2.04788×
-6
10
,1.69358×
-6
10
,1.35817×
-6
10
,1.04527×
-6
10
,7.55139×
-7
10
,4.86052×
-7
10
,2.35345×
-7
10
,0.},{0.,4.76121×
-7
10
,9.00212×
-7
10
,1.25173×
-6
10
,1.51675×
-6
10
,1.68888×
-6
10
,1.76936×
-6
10
,1.76644×
-6
10
,1.69393×
-6
10
,1.56904×
-6
10
,1.40978×
-6
10
,1.23243×
-6
10
,1.04974×
-6
10
,8.70372×
-7
10
,6.99363×
-7
10
,5.39011×
-7
10
,3.89781×
-7
10
,2.51044×
-7
10
,1.21597×
-7
10
,0.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.}}
Transpose[%273]
As we can see from above, all the difference between 189th recurrsion and 190th recurrsion are smaller than 10^(-5), which means the result converges enough.
In[]:=
AbsoluteTiming[Last[NestList[pm,{{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,10.,9.,8.,7.,6.,5.,4.,3.,2.,0.},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}},189]]]