BeginPackage["Schofield`"] $Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0} β[D_:2] := Piecewise[ { {1/8, D == 2}, {0.326419, D == 3}, {1/2, D == 4}, {β, True} } ] δ[D_:2] := Piecewise[ { {15, D == 2}, {4.78984, D == 3}, {3, D == 4}, {δ, True} } ] α[D_:2] := Piecewise[ { {0, D == 2}, {0.11008, D == 3}, {0, D == 4}, {α, True} } ] Δ[D_:2] := β[D] δ[D] OverBar[s] := 2^(1/12) Exp[-1/8] Glaisher^(3/2) t[θ_] := ((θ/1)^2 - 1) h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i] θ^(2*i+1), {i, 0, n}] η[g_][θ_] := t[θ] / (g[θ] / I)^(1 / Δ[2]) RFLow[B_, θc_][θ_] := (1/\[Pi])(2 E^(1/( B \[Theta]c)) \[Theta]c ExpIntegralEi[-(1/(B \[Theta]c))] + E^(1/(B (-\[Theta] + \[Theta]c))) (\[Theta] - \[Theta]c) \ ExpIntegralEi[1/(B \[Theta] - B \[Theta]c)] - E^(1/(B \[Theta] + B \[Theta]c)) (\[Theta] + \[Theta]c) ExpIntegralEi[-(1/( B \[Theta] + B \[Theta]c))]) RFHigh[ξ0_][ξ_] := (ξ^2+ξ0^2)^(5/6) - (ξ0^2)^(5/6) RF[n_][θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + Sum[A[i] θ^(2 i), {i, 1, n}] RFReg[n_][θ_] := AL (1/\[Pi])(2 E^(1/( B \[Theta]c)) \[Theta]c ExpIntegralEi[-(1/(B \[Theta]c))] - E^(1/(B \[Theta] + B \[Theta]c)) (\[Theta] + \[Theta]c) ExpIntegralEi[-(1/( B \[Theta] + B \[Theta]c))]) + AH RFHigh[θ0][θ] + Sum[A[i] θ^(2 i), {i, 1, n}] dRFc[n_][m_] := Piecewise[{{AL m! Gamma[m - 1] B^(m - 1) / π, m>1}, {0, True}}] + D[RFReg[n][θ], {θ, m}] /. θ -> θc RFC[n_][θ_] := RF[n][θ] + AL I Sign[Im[θ]] ((θ-θc)Exp[-1/(B(θ-θc))]-(-θ-θc)Exp[-1/(B(-θ-θc))]) ddξ[h_][f_] := D[f, θ] / D[h[θ] / RealAbs[t[θ]]^Δ[2], θ] ddη[h_][f_] := D[f, θ] / D[t[θ] / h[θ]^(1 / Δ[2]), θ] invDerivativeList[n_][f_][x_] := Module[ {xp, dfs, fp, Pns}, dfs = Rest[NestList[D[#, xp] &, f[xp], n]] /. xp -> x; Pns = FoldList[{Pm, m} |-> fp'[xp] D[Pm, xp] - (2 m - 1) fp''[xp] Pm, 1, Range[n - 1]] /. Derivative[m_][fp][xp] :> dfs[[m]]; MapIndexed[{Pn, i} |-> Pn/dfs[[1]]^(2 i[[1]] - 1), Pns] ] dFdξLowList[n_, h_][m_] := Module[ { ds, dF, df }, ds = invDerivativeList[m+1][Function[θ, h[θ] / t[θ]^Δ[2]]][θc]; dF = NestList[Function[f, D[f, θ]], RFReg[n][θ], m] + Table[Piecewise[{{AL k! Gamma[k - 1] B^(k - 1)/\[Pi], k > 1}, {0, True}}], {k, 0, m}] /. θ -> θc; df = NestList[D[#, \[Theta]] &, fp[\[Theta]]/t[\[Theta]]^2 - 1/(8 \[Pi]) Log[t[\[Theta]]^2], m] /. Map[Derivative[#][fp][\[Theta]] -> dF[[# + 1]] &, Range[0, m]] /. θ -> θc; Table[Sum[df[[k+1]] BellY[j, k, ds[[;; j - k + 1]]], {k, 0, j}]/(j!), {j, 0, m}] ] dFdξHighList[n_, h_][m_] := Module[ { ds, dF, df }, ds = invDerivativeList[m+1][Function[θ, h[θ] / (-t[θ])^Δ[2]]][0]; dF = NestList[Function[f, D[f, θ]], RF[n][θ], m] /. θ -> 0; df = NestList[D[#, \[Theta]] &, fp[\[Theta]]/t[\[Theta]]^2 - 1/(8 \[Pi]) Log[t[\[Theta]]^2], m] /. Map[Derivative[#][fp][\[Theta]] -> dF[[# + 1]] &, Range[0, m]] /. θ -> 0; Table[Sum[df[[k+1]] BellY[j, k, ds[[;; j - k + 1]]], {k, 0, j}]/(j!), {j, 0, m}] ] dFdξLow[n_, h_][m_] := Module[{ff, hh}, Nest[ddξ[hh], ff[θ] / t[θ]^2 - Log[t[θ]^2] / (8 π), m] /. θ -> θc /. Map[Derivative[#][ff][θc] -> dRFc[n][#] &, Range[0, m]] /. hh -> h] dFdξHigh[n_, h_][m_] := Module[{ff, hh}, Nest[ddξ[hh], ff[θ] / t[θ]^2 - Log[t[θ]^2] / (8 π), m] /. θ -> 0 /. Map[Derivative[#][ff][0] -> eqHighRHS[RF[n]][#] &, Range[0, m]] /. hh -> h] dFdη[n_, h_][m_][tt_] := Module[{ff, hh}, Nest[ddη[hh], h[θ]^(-2 / Δ[]) (ff[θ] - t[θ]^2 Log[hh[θ]^2] / (8 π Δ[])), m] /. θ -> tt /. Map[Derivative[#][ff][tt] -> Derivative[#][RF[n]][tt] &, Range[0, m]] /. hh -> h] dFdηList[n_, h_][m_][tt_] := Module[{ff, hh}, NestList[ddη[hh], h[θ]^(-2 / Δ[2]) (ff[θ] - t[θ]^2 Log[hh[θ]^2] / (8 π Δ[2])), m] /. θ -> tt /. Map[Derivative[#][ff][tt] -> Derivative[#][RF[n]][tt] &, Range[0, m]] /. hh -> h] ruleB[g_] := B -> (2 * OverBar[s] / π) * (- g'[θc] / t[θc]^Δ[2]) ruleθ0[g_] := Around[0.18930, 0.00005] - Simplify[g[I θ0]/(-t[I θ0])^Δ[2]/I] ruleAL[g_] := AL -> Exp[Δ[2] t[θc]^(Δ[2] - 1) t'[θc] / (2 OverBar[s] / π g'[θc]) - t[θc]^Δ[2] g''[θc] / (4 OverBar[s] / π g'[θc]^2)] t[θc]^(1/8) OverBar[s] / (2 π) * g'[θc] ruleAH[g_] := AH / ((g[I θ0]/ I)^(2 / Δ[2]) * (-η[g]'[I θ0] / (2 θ0 I))^(5/6)) + Around[1.37, 0.02] eqLowRHSReg[n_][m_] := dRFc[n][m] eqLowLHS[h_][m_] :=D[ t[θ]^2 (Gl[h[θ] t[θ]^-Δ[2]] + Log[t[θ]^2]/(8 π)), {θ, m} ] /. θ -> θc eqLow[n_, h_][m_] := (eqLowRHSReg[n][m] - eqLowLHS[h][m]) / m! eqHighRHS[F_][m_] := D[F[θ], {θ, m} ] /. θ -> 0 eqHighLHS[h_][m_] := D[(-t[θ])^2 (Gh[h[θ] (-t[θ])^-Δ[2]] + Log[(-t[θ])^2]/(8 π)), {θ, m} ] /. θ -> 0 eqHigh[n_, h_][m_] := (eqHighRHS[RF[n]][m] - eqHighLHS[h][m]) / m! eqMid[F_, h_][m_] := D[ F[θ] - t[θ]^2 Log[h[θ]^2]/(8 Δ[2]π) - h[θ]^((2-α[2])/Δ[2]) Φ[η] /. η -> t[θ] / h[θ]^(1 / Δ[2]), {θ, m} ] / m! /. θ -> 1 δ0 = 10^(-14); Φs = { -1.19773338379799339, -0.31881012489061, 0.110886196683, 0.01642689465, -2.639978 10^-4, -5.140526 10^-4, 2.08856 10^-4, -4.4819 10^-5, 3.16 10^-7, 4.31 10^-6, -1.99 10^-6 } Gls = { 0, -OverBar[s], −1.000960328725262189480934955172097320572505951770117 Sqrt[2]/((2 )^(-7/8) (2^(3/16)/OverBar[s])^2)/2/(12 \[Pi]), Around[ 0.038863932, 3.0 10^(-9)], Around[−0.068362119, 2.0 10^(-9)], Around[ 0.18388371, 1.0 10^(-8)], Around[-0.659170, 1.0 10^(-6)], Around[ 2.937665, 3.0 10^(-6)], Around[-15.61, 1.0 10^(-2)], Around[ 96.76, 1.0 10^(-2)], -6.79 10^2, 5.34 10^3, -4.66 10^4, 4.46 10^5, -4.66 10^6 } Ghs = { 0, 0, -1.000815260440212647119476363047210236937534925597789 Sqrt[2]/((2 )^(-7/8) (2^(3/16)/OverBar[s])^2)/2, 0, Around[ 8.333711750, 5.0 10^(-9)], 0, Around[-95.16897, 3.0 10^(-5)], 0, Around[1457.62, 3.0 10^(-2)], 0, Around[-2.5891 10^4, 2.0], 0, 5.02 10^5, 0, -1.04 10^7 } dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)! ΦRules = MapIndexed[dRule[Φ], Φs]; GlRules = MapIndexed[dRule[Gl], Gls]; GhRules = MapIndexed[dRule[Gh], Ghs]; ClearAll[rules] rules[g_] := Join[ΦRules, GlRules, GhRules, {ruleAL[g], ruleB[g], gC[0]->1}] eq[n_, g_][m_, p_, q_] := Flatten[Join[{ruleθ0[g], ruleAH[g], g'[0] θc - 1}, eqLow[n, g][#] & /@ Range[0, m],eqMid[RF[n], g][#] & /@ Range[0, p], eqHigh[n, g] /@ Range[2, q, 2]]] //. rules[g] /. Around[x_, _] :> x eqAround[n_, g_][m_, p_, q_] := Flatten[Join[{ruleθ0[g], ruleAH[g]}, eqLow[n, g][#] & /@ Range[0, m],eqMid[RF[n], g][#] & /@ Range[0, p], eqHigh[n, g] /@ Range[2, q, 2]]] //. rules[g] formResiduals[data_, functions_, δ_:10^(-15)] := If[Head[#1]===Around, (#1["Value"] - #2) / Max[#1["Uncertainty"], δ], (#1 - #2) / δ] & @@@ Thread[{data, functions}] resLow[n_, g_, δ_][m_] := formResiduals[Gls[[;;m+1]], dFdξLowList[n, g][m], δ] resHigh[n_, g_, δ_][m_] := Rest[formResiduals[Ghs[[;;m+1]], dFdξHighList[n, g][m], δ][[;;;;2]]] res[F_, g_, δ_][m_] := Join[resLow[F, g, δ][m], resHigh[F, g, δ][m], {ruleθ0[g] / 0.00005, ruleAH[g] / 0.02} /. Around[x_, _] :> x] chiSquared[F_, g_, δ_][m_] := Total[res[F, g, δ][m]^2] newSol[eqs_, oldSol_, newVars_, δ_:0, γ_:0, opts___] := FindRoot[ eqs, Join[ {#1, #2 + γ * RandomVariate[NormalDistribution[]]} & @@@ (oldSol /. Rule -> List), Thread[{newVars, δ * RandomVariate[NormalDistribution[], Length[newVars]]}] ], MaxIterations -> 500, opts ] levenbergMarquardtStep[M_, λ_, g_] := LinearSolve[M + λ DiagonalMatrix[Diagonal[M]], g] levenburgMarquardt[r_, β0_, λ0_ : 1, ν_ : 2] := Module[ { n = Length[β0], β = β0, λ = λ0, newβ = β0, x, rf, Jf, oldJ, oldr, newr, M, g, δ, oldC, newC }, PrintTemporary["Compiling the Jacobian..."]; rf = Compile[{{x, _Real, 1}}, Evaluate[r /. Thread[Rule[First /@ β, Part[x, #] & /@ Range[n]]]] ]; Jf = Compile[{{x, _Real, 1}}, Evaluate[D[r, {First /@ β}] /. Thread[Rule[First /@ β, Part[x, #] & /@ Range[n]]]] ]; PrintTemporary["Beginning the algorithm."]; oldr = rf[β[[All, 2]]]; oldJ = Jf[β[[All, 2]]]; oldC = Re[Total[oldr^2]]; g = Re[Transpose[oldJ] . oldr]; M = Re[Transpose[oldJ] . oldJ]; δ = levenbergMarquardtStep[M, λ / ν, g]; PrintTemporary[Dynamic[oldC]] While[Norm[δ] > 10^-15, newβ[[All, 2]] -= δ; newr = rf[newβ[[All, 2]]]; newC = Re[Total[newr^2]]; While[newC > oldC, δ = levenbergMarquardtStep[M, λ, g]; newβ = β; newβ[[All, 2]] -= δ; newr = rf[newβ[[All, 2]]]; newC = Re[Total[newr^2]]; λ *= ν; ]; λ /= ν; oldC = newC; oldr = newr; β = newβ; oldJ = Jf[β[[All, 2]]]; g = Re[Transpose[oldJ] . oldr]; M = Re[Transpose[oldJ] . oldJ]; δ = levenbergMarquardtStep[M, λ / ν, g]; ]; {newC, β} ] EndPackage[]