Much work.

1 files changed, 29 insertions, 17 deletions

diff --git a/schofield.wl b/schofield.wl
index b26b662..7f6bed8 100644
--- a/schofield.wl
+++ b/schofield.wl
@@ -39,9 +39,9 @@ t[θ_] := ((θ)^2 - 1)
 hBasis = Orthogonalize[x^# & /@ Range[0, 20], Function[{f, g}, Integrate[f g (1 - x^2)^2, {x, -1, 1}]]]
 
 h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i] hBasis[[2 * i + 2]], {i, 0, n}] /. x -> θ / θc
-h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]θ^(2 * i + 1), {i, 0, n}]
-*)
 h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]LegendreP[(2 * i + 1), θ/θc], {i, 0, n}]
+*)
+h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]θ^(2 * i + 1), {i, 0, n}]
 
 RFLow[B_, θc_][θ_] := (1/\[Pi])(2 E^(1/(
    B \[Theta]c)) \[Theta]c ExpIntegralEi[-(1/(B \[Theta]c))] + 
@@ -52,35 +52,44 @@ ExpIntegralEi[1/(B \[Theta] - B \[Theta]c)] -
      B \[Theta] + B \[Theta]c))])
 RFHigh[ξ0_][ξ_] := (ξ^2+ξ0^2)^(5/6)
 
-RF[n_][θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + Sum[A[i] LegendreP[2i, θ], {i, 0, n}]
+RF[n_][θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + Sum[A[i] θ^(2 i), {i, 0, 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] LegendreP[2i, θ], {i, 0, 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]
+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]
+
 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 + t[θc]^2 OverBar[s] / (2 π) * (- g'[θc] / t[θc]^Δ[2])
-ruleg0[g_] := 1 - g'[0]
 
-eqLowRHS[F_][m_] := SeriesCoefficient[F[θ], {θ, θc, m}, Assumptions -> Join[{θ < θc, θ > 1}, $Assumptions]]
+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} ] / m! /. θ -> θc
+  {θ, m} ] /. θ -> θc
 
-eqLow[F_, h_][m_] := eqLowRHS[F][m] - eqLowLHS[h][m]
+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[F_, h_][m_] := eqHighRHS[F][m] - eqHighLHS[h][m]
+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} ] /. θ -> 1
 
-δ0 = 10^-16;
+δ0 = 10^-14;
 
 Φs = {
   -1.197733383797993,
@@ -109,14 +118,14 @@ Gls = {
 }
 
 Ghs = {
-  0,
-  0,
+  Around[0, δ0],
+  Around[0, δ0],
   Around[ -1.84522807823, 10^(-11)],
-  0,
+  Around[0, δ0],
   Around[  8.3337117508, 10^(-10)],
-  0,
+  Around[0, δ0],
   Around[-95.16897, 10^(-5)],
-  0,
+  Around[0, δ0],
   Around[1457.62, 3 10^(-2)],
   0,
   Around[-25891, 2],
@@ -134,11 +143,14 @@ rules := Join[ΦRules, GlRules, GhRules]
 (*ξ0 := 0.18930*)
 (*gC[0] := 1*)
 tC[0] := 1
+gC[0] := 1
 
-eq[F_, g_][m_] := Flatten[Join[{ruleg0[g], ruleB[g], ruleθ0[g]},{eqLow[F, g][#](*, eqMid[F, g][#]*)} & /@ Range[0, m], eqHigh[F, g] /@ Range[0, m, 2]]] //. rules /. Around[x_, _] :> x
+eq[n_, g_][m_] := Flatten[Join[{ruleB[g], ruleθ0[g]},{eqLow[n, g][#](*, eqMid[F, g][#]*)} & /@ Range[0, m], eqHigh[n, g] /@ Range[0, m, 2]]] //. rules /. Around[x_, _] :> x
 
-chiSquaredLow[F_, g_][m_] := Total[(((#[[1]] /. rules)["Value"] - #[[2]])^2 / (#[[1]] /. rules)["Uncertainty"]^2)& /@ Solve[0 == (eqLow[ff, g] /@ Range[0, m]), Derivative[#][Gl][0]& /@ Range[0, m]][[1]]] /. ff -> F
-chiSquaredHigh[F_, g_][m_] := Total[(((#[[1]] /. rules)["Value"] - #[[2]])^2 / (#[[1]] /. rules)["Uncertainty"]^2)& /@ Solve[0 == (eqHigh[ff, g] /@ Range[2, m, 2]) /. ff'[0] -> 0, Derivative[#][Gh][0]& /@ Range[2, m, 2]][[1]]] /. ff -> F
+ (* *)
+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,