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BeginPackage["Schofield`"]
$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 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] := 1.357838341706595496
t[θ_] := ((θ)^2 - 1)
h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]LegendreP[(2 * i + 1), θ/θc], {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)
RF[n_][θ_] := AL RFLow[B, θc][θ] RFHigh[θ0][θ] + Sum[A[i] LegendreP[(2 i), θ/θc] , {i, 1, n}]
RFReg[n_][θ_] := AL RFHigh[θ0][θ] (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))]) + Sum[A[i] LegendreP[(2 i), θ/θc], {i, 1, n}]
dRFc[n_][m_] := AL m! Sum[Piecewise[{{ Gamma[j - 1] B^(j - 1) / π, j>1}, {0, True}}] (D[RFHigh[θ0][θ], {θ, m - j}] / (m - j)! /. θ -> θc), {j, 0, m}] + 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]
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]
dFdξLowList[n_, h_][m_] := Module[{ff, hh}, NestList[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]] /. Map[Derivative[#][hh][θc] -> Derivative[#][h][θc] &, Range[0, m]]]
dFdξHighList[n_, h_][m_] := Module[{ff, hh}, NestList[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]
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_] := Simplify[g[I θ0]/(-t[I θ0])^Δ[2]/I] - 0.18930
ruleAL[g_] := AL RFHigh[θ0][θc] + t[θc]^2 OverBar[s] / (2 π) * (- g'[θc] / t[θc]^Δ[2])
ruleAH[g_] := AL Re[RFLow[B, θc][θ0 I]]+ 1.37 * (g[I θ0]/ I)^(2 / Δ[2]) * (-η[g]'[I θ0] / (2 θ0 I))^(5/6)
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 = {
-1.197733383797993,
-0.318810124891,
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 = {
Around[0, δ0],
Around[-OverBar[s], δ0],
Around[−0.048953289720, 2 10^(-12)],
Around[ 0.0388639290, 1 10^(-10)],
Around[-0.068362121, 1 10^(-9)],
Around[ 0.18388371, 1 10^(-8)],
Around[-0.659170, 1 10^(-6)],
Around[ 2.937665, 3 10^(-6)],
Around[-15.61, 10^(-2)],
96.76,
-6.79 10^2,
5.34 10^3,
-4.66 10^4,
4.46 10^5,
-4.66 10^6
}
Ghs = {
Around[0, δ0],
Around[0, δ0],
Around[ -1.84522807823, 10^(-11)],
Around[0, δ0],
Around[ 8.3337117508, 10^(-10)],
Around[0, δ0],
Around[-95.16897, 10^(-5)],
Around[0, δ0],
Around[1457.62, 3 10^(-2)],
0,
Around[-2.5891 10^4, 2],
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[gC]
rules := Join[ΦRules, GlRules, GhRules]
(*ξ0 := 0.18930*)
(*gC[0] := 1*)
tC[0] := 1
(*gC[0] := 1*)
eq[n_, g_][m_, p_, q_] := Flatten[Join[{ruleB[g], ruleθ0[g], g'[0] - 1}, eqLow[n, g][#] & /@ Range[0, m],eqMid[RF[n], g][#] & /@ Range[0, p], eqHigh[n, g] /@ Range[2, q, 2]]] //. rules /. Around[x_, _] :> x
(* *)
chiSquaredLow[n_, g_][m_] := Total[(((#[[1]] /. rules)["Value"] - #[[2]])^2 / (#[[1]] /. rules)["Uncertainty"]^2)& /@ ({Gls[[#+1]], dFdξLow[n, g][#] / #!} & /@ Range[0, m])]
chiSquaredHigh[n_, g_][m_] := Total[(((#[[1]] /. rules)["Value"] - #[[2]])^2 / (#[[1]] /. rules)["Uncertainty"]^2)& /@ ({Ghs[[#+1]], dFdξHigh[n, g][#] / #!} & /@ Range[0, m])]
chiSquared[F_, g_][m_] := chiSquaredLow[F, g][m] + chiSquaredHigh[F, g][m] + ruleB[g]^2 / δ0^2 + ruleθ0[g]^2 / 0.00005^2
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 -> 50000,
opts
]
EndPackage[]
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