Get["~/doc/research/first_order_singularities/mma/schofield.wl"]

SetAttributes[untilSuccess,HoldAll];
untilSuccess[e_]:=Quiet[Check[e,untilSuccess[e]]]

randomEval[eq_]:=eq/.Around[x_,y_]:>RandomVariate[NormalDistribution[x,y]]

$DefaultRemoteKernel=RemoteKernelObject["ssh://math@sident.lassp.cornell.edu"];

Match ends

Generate fits

nAs=2;

s0=#1->Re[#2]&@@@untilSuccess[newSol[randomEval[eqAround[2,h[0]][1,-1,2]],{θc->1+RandomReal[],θ0->0.2},{AH,A[1],A[2]},.1]]

Timing[dGdξLowList[nAs,h[0]][6]/(Range[0,6]!)//.rules[h[0]]/.s0]

ListLogPlot[{
Abs[Gls],
Chop[Abs[dGdξLowList[2,h[0]][8]]//.rules[h[0]]/.S[2]]
}]

ListLogPlot[{
Abs[Ghs],
Chop[Abs@dGdξList[2,h[0]][10,0]//.rules[h[0]]/.S[2]]
}]

m0=annealFit[res[2,h[0],(10^(-12))][2],s0]

m1=annealFit[res[2,h[1],(10^(-12))][3],Join[m0[[2]],{gC[1]->0}]]

m2=annealFit[res[3,h[2],(10^(-12))][4],Join[m1[[2]],{gC[2]->0,A[3]->0}]]

m3=annealFit[res[3,h[3],(10^(-12))][5],Join[m2[[2]],{gC[3]->0}],(10^(-2))]

m4=annealFit[res[4,h[4],(10^(-12))][6],Join[m3[[2]],{gC[4]->0,A[4]->0}],(10^(-3))]

m5=RemoteEvaluate@annealFit[res[4,h[5],(10^(-10))][7],Join[m4[[2]],{gC[5]->0}]]

m6=RemoteEvaluate@annealFit[res[5,h[6],(10^(-6))][8],Join[m5[[2]],{gC[6]->0,A[5]->0}],(10^(-3))]

m61=RemoteEvaluate@annealFit[res[5,h[6],(10^(-10))][8],m6[[2]]]

m6=RemoteEvaluate@annealFit[res[5,h[6],(10^(-6))][8],m7[[2,;;-2]]]

m61=RemoteEvaluate@annealFit[res[5,h[6],(10^(-10))][8],m6[[2]]]

m7=RemoteEvaluate@annealFit[res[5,h[7],(10^(-5))][9],Join[m61[[2]],{gC[7]->0}],(10^(-2))]

m7

m71=RemoteEvaluate@annealFit[res[5,h[7],(10^(-10))][9],m7[[2]]]

m8=RemoteEvaluate@annealFit[res[6,h[8],(10^(-5))][10],Join[m7[[2]],{gC[8]->0,A[6]->0}],(10^(-3))]

m81=RemoteEvaluate@annealFit[res[6,h[8],(10^(-10))][10],Join[m8[[2]],{}]]

m9=RemoteEvaluate@annealFit[res[6,h[9],δ][11]//.rules[h[9]],δ,Join[m8[[2]],{gC[9]->0}]]

m81=Import["~/doc/research/first_order_singularities/mma/m8.wl"]

m81=FindRoot[Evaluate[D[Total[((res[6,h[8],(10^(-15))][10])^2)]//.rules[h[8]],{m8[[2,All,1]]}]],Evaluate[m8[[2]]/.Rule->List],WorkingPrecision->20,MaxIterations->5000]

test=((Φs[[;;3+1]]-#)(*/Φs[[;;ntest+1]]*))&@dΦdηList[NF[#],H[#]][3,1]//.rules[H[#]]/.S[#]&/@Range[2,11];

pCov=ListLogPlot[Thread[{Range[2,11],#}]&/@Abs@Re@Transpose[test],Joined->True,
AxesLabel->{"Order","((Subscript[Δℱ, 0])^((n)))"},PlotLegends->LineLegend[{0,1,2,3},LegendLabel->n]]

Export["/tmp/convergence.pdf",pCov];

ListLogPlot[(Abs@ruleθ0[H[#]]/.S[#]/.gC[0]->1)&/@Range[2,11],PlotRange->{(10^(-6)),.1}]

ListPlot[(Re@ruleAH[H[#]]/.S[#]/.gC[0]->1)&/@Range[2,11]]

ListLogLogPlot[(n|->Table[{1/(2i+1),Abs@((SeriesCoefficient[H[n][θ],{θ,0,2i+1}])/(SeriesCoefficient[H[n][θ],{θ,0,2i-1}]))},{i,1,11}]/.S[n]/.gC[0]->1)/@Range[6,11],Joined->True]

ListLogLogPlot[Table[{1/(2i),Abs@((A[i])/(A[i-1]))},{i,1,10}]/.(S/@Range[8,10]),Joined->True]

Plot[Evaluate[H[#][θ ]/.gC[0]->1/.S[#]&/@Range[8,10]],{θ,0,1.4}]

Plot[Evaluate[ℱ[#][θ ]//.rules[H[#]]/.S[#]&/@Range[8,10]],{θ,0,1.4}]

ParametricPlot[Evaluate[{-t[θ]((H[#][θ ])^(-1/Δ[])),((H[#][θ])^(-2/Δ[]))(ℱ[#][θ ]-((t[θ])^2)((Log[((H[#][θ])^2)])/(8 π Δ[])))-(((t[θ]((H[#][θ ])^(-1/Δ[]))))^2)Log[(((t[θ]((H[#][θ ])^(-1/Δ[]))))^2)](1/(8π))}//.rules[H[#]]/.S[#]&/@Range[8,10]],{θ,0.00001,1.4},AspectRatio->1/GoldenRatio,PlotRange->{{-50,50},{-2.5,0.5}},PlotPoints->400]

ParametricPlot[Evaluate[{H[#][θ]((Abs[t[θ]])^(-Δ[])),((t[θ])^(-2))ℱ[#][θ ]-((Log[((t[θ])^2)])/(8 π ))}//.rules[H[#]]/.S[#]&/@Range[7,10]],{θ,1,2},AspectRatio->1/GoldenRatio]

ParametricPlot[Evaluate[{H[#][θ]((Abs[t[θ]])^(-Δ[])),((t[θ])^(-2))ℱ[#][θ ]-((Log[((t[θ])^2)])/(8 π ))}//.rules[H[#]]/.S[#]&/@Range[7,10]],{θ,-0.8,0.8},AspectRatio->1/GoldenRatio]

complexField=1.8;ComplexPlot[RFC[NF[9]][θ]//.rules[H[9]]/.S[9],{θ,-complexField-complexField/GoldenRatio  I,complexField+complexField/GoldenRatio I},PlotPoints->200,Exclusions->None,ColorFunction->{Automatic,"None"}]

ComplexPlot[Evaluate[RFC[0][θ]/.AH->0/.{B->1,θc->1,AL->1}],{θ,1-0.05-0.05I,1+0.05+0.05I},Exclusions->None,ColorFunction->{"Automatic",None},PlotPoints->100,WorkingPrecision->500]

complexField=0.2;ComplexPlot[RFC[NF[9]][θ]//.rules[H[9]]/.S[9],{θ,θc-complexField-complexField/GoldenRatio  I,θc+complexField+complexField/GoldenRatio I}/.S[9],PlotPoints->100,Exclusions->None,ColorFunction->{Automatic,"None"},WorkingPrecision->200]

Plot[Evaluate[Arg@RFC[0][θ+I (10^(-7))]/.AH->0/.{B->1,θc->1,AL->1}],{θ,0.9,1.1},WorkingPrecision->200,PlotPoints->50]

Stored fits

H[2]=h[0];
NF[2]=2;
ℱ[2]=RF[2];
S[2]={θc->1.2113984981855266`,θ0->0.18040968433960122`,AH->-2.1295236147887016`,A[1]->1.2447338967832837`,A[2]->0.4997458274069301`};

H[3]=h[1];
NF[3]=2;
ℱ[3]=RF[2];
S[3]={θc->1.3497610103287452`,θ0->0.19612583387698465`,AH->-2.199941956875278`,A[1]->1.2401996118460512`,A[2]->0.39028245506270876`,gC[1]->-0.014909109302639726`};

H[4]=h[2];
NF[4]=3;
ℱ[4]=RF[3];
S[4]={θc->1.449037399770833`,θ0->0.1956310634797022`,AH->-2.2433174743125224`,A[1]->1.2964274810848326`,A[2]->0.3707956103394163`,gC[1]->-0.10871082597145154`,gC[2]->-0.0031747260088083834`,A[3]->-0.028926044331647412`};

H[5]=h[3];
NF[5]=3;
ℱ[5]=RF[3];
S[5]={θc->1.4718997870644017`,θ0->0.19553388787373974`,AH->-2.2320560452062117`,A[1]->1.2804295612857017`,A[2]->0.3631783861327246`,gC[1]->-0.11398650090024323`,gC[2]->-0.0031668813000029347`,A[3]->-0.0289243859800091`,gC[3]->8.857419924924222`*^-7};

H[6]=h[4];
NF[6]=4;
ℱ[6]=RF[4];
S[6]={θc->1.435757396344316`,θ0->0.19736542866520598`,AH->-2.3981472287670393`,A[1]->1.5091177600111922`,A[2]->0.39200424741821555`,gC[1]->-0.19533215126551273`,gC[2]->0.029301113923157842`,A[3]->-0.09002270806803321`,gC[3]->0.003990623031590741`,gC[4]->-0.00011912993661574494`,A[4]->0.01723324643602473`};

H[7]=h[5];
NF[7]=4;
ℱ[7]=RF[4];
S[7]={θc->1.4323668647862395`,θ0->0.19730236197017473`,AH->-2.42285516121067`,A[1]->1.5453691206248084`,A[2]->0.4131963567936482`,gC[1]->-0.22076846011611728`,gC[2]->0.036245425139259865`,A[3]->-0.12160894976267282`,gC[3]->0.010119794698736752`,gC[4]->-0.001143426002373578`,A[4]->0.026345666471523435`,gC[5]->0.00010094806835660409`};

H[8]=h[6];
NF[8]=5;
ℱ[8]=RF[5];
S[8]={θc->1.3710192066561966`,θ0->0.19655268865457023`,AH->-2.551315962883039`,A[1]->1.732278814614242`,A[2]->0.5967868834184186`,gC[1]->-0.3515062191585261`,gC[2]->0.005023244859308164`,A[3]->-0.2952087119338217`,gC[3]->0.05365943568010773`,gC[4]->-0.01980645818232528`,A[4]->0.0785090329663038`,gC[5]->0.003353070367769979`,gC[6]->-0.00026033616168888695`,A[5]->-0.007251366969801466`};

H[9]=h[7];
NF[9]=5;
ℱ[9]=RF[5];
S[9]={θc->1.375435411428919`,θ0->0.1965201354188921`,AH->-2.548202398429163`,A[1]->1.7277523989023333`,A[2]->0.602777511249843`,gC[1]->-0.35476072551695503`,gC[2]->-0.0071698284978869015`,A[3]->-0.28737467546972323`,gC[3]->0.052493450595160036`,gC[4]->-0.019894421937458582`,A[4]->0.07241135663444981`,gC[5]->0.003534527038057842`,gC[6]->-0.00033818931210095453`,A[5]->-0.007245524506332198`,gC[7]->0.000010619260978013524`};

H[10]=h[8];
NF[10]=6;
ℱ[10]=RF[6];
S[10]={θc->1.3945095399721226`,θ0->0.19657331402448114`,AH->-2.540815351266595`,A[1]->1.7168304596304513`,A[2]->0.58914437918972`,gC[1]->-0.3578933157155564`,gC[2]->-0.0002471276399885635`,A[3]->-0.28467214257786666`,gC[3]->0.05289863457865093`,gC[4]->-0.0201090322678147`,A[4]->0.073168474426385`,gC[5]->0.0034486104788930307`,gC[6]->-0.00021837125394706086`,A[5]->-0.007098620515609658`,gC[7]->-0.000025781921631023817`,gC[8]->3.737427990965571`*^-6,A[6]->0.000017492654253389754`};

H[11]=h[9];
NF[11]=6;
ℱ[11]=RF[NF[11]];
S[11]=Import["~/doc/research/first_order_singularities/mma/m11.wl"][[2]]