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-rw-r--r--new_schofield.nb19414
-rw-r--r--schofield.wl79
2 files changed, 14169 insertions, 5324 deletions
diff --git a/new_schofield.nb b/new_schofield.nb
index 97284a1..86da5b9 100644
--- a/new_schofield.nb
+++ b/new_schofield.nb
@@ -10,10 +10,10 @@
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+ Annotation[#, "Charting`Private`Tag$26063#1"]& ]}, {}},
+ AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
+ Axes->{True, True},
+ AxesLabel->{None, None},
+ AxesOrigin->{0, 0},
+ DisplayFunction->Identity,
+ Frame->{{False, False}, {False, False}},
+ FrameLabel->{{None, None}, {None, None}},
+ FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
+ GridLines->{None, None},
+ GridLinesStyle->Directive[
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+ ImagePadding->{{18.28172831632653, 0.37500000000005684`}, {
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+ Method->{
+ "DefaultBoundaryStyle" -> Automatic,
+ "DefaultGraphicsInteraction" -> {
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+ "Effects" -> {
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+ "Droplines" -> {
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+ "placement" -> {"x" -> "All", "y" -> "None"}}}},
+ "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" ->
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+ (Identity[#]& )[
+ Part[#, 1]],
+ (Identity[#]& )[
+ Part[#, 2]]}& ), "CopiedValueFunction" -> ({
+ (Identity[#]& )[
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+ (Identity[#]& )[
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-Cell[2287715, 44881, 491, 10, 24, "Input",ExpressionUUID->"8858ddfc-1e71-4778-a0b8-512abba3d901"],
-Cell[2288209, 44893, 22737, 390, 179, "Output",ExpressionUUID->"3a7e88f8-603a-4c66-8a1b-065c43aff3cd"]
+Cell[2733091, 53557, 491, 10, 24, "Input",ExpressionUUID->"8858ddfc-1e71-4778-a0b8-512abba3d901"],
+Cell[2733585, 53569, 22737, 390, 179, "Output",ExpressionUUID->"3a7e88f8-603a-4c66-8a1b-065c43aff3cd"]
}, Open ]],
-Cell[2310961, 45286, 3849, 95, 75, "Input",ExpressionUUID->"954f68de-17bb-4b28-b089-18f4c1840d53"],
+Cell[2756337, 53962, 3849, 95, 75, "Input",ExpressionUUID->"954f68de-17bb-4b28-b089-18f4c1840d53"],
Cell[CellGroupData[{
-Cell[2314835, 45385, 448, 10, 24, "Input",ExpressionUUID->"fd8d9fbd-6567-43cc-a4dc-4c986b02e466"],
-Cell[2315286, 45397, 26854, 485, 178, "Output",ExpressionUUID->"b7f39439-0c1d-40f8-af40-5fed6fd931de"]
+Cell[2760211, 54061, 448, 10, 24, "Input",ExpressionUUID->"fd8d9fbd-6567-43cc-a4dc-4c986b02e466"],
+Cell[2760662, 54073, 26854, 485, 178, "Output",ExpressionUUID->"b7f39439-0c1d-40f8-af40-5fed6fd931de"]
}, Open ]]
}, Open ]]
}
diff --git a/schofield.wl b/schofield.wl
index 646881d..8ac7ffc 100644
--- a/schofield.wl
+++ b/schofield.wl
@@ -1,7 +1,7 @@
BeginPackage["Schofield`"]
-$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0}
+$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 0}
β[D_:2] := Piecewise[
{
@@ -31,28 +31,43 @@ $Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0}
]
Δ[D_:2] := β[D] δ[D]
-t[θ_] := (θ)^2 - 1
-h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i] θ^(2i+1), {i, 0, n}]
-
-RFf[y_] := π^-1 y Exp[1/y] ExpIntegralEi[-1/y]
-RFnonsingular[n_][θ_] := A RFf[B(θc+θ)] + Sum[FC[i] θ^(2i) , {i, 0, n}]
-RF[n_][θ_] := A RFf[B(θc-θ)] + RFnonsingular[n][θ]
+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}]
+
+RFLow[B_, θc_][θ_] := (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_] := (1+(ξ/ξ0)^2)^(5/6) - 1
+
+RF[θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + AM θ^2
+
+ruleB[g_] := B - π / (2 1.3578383417066) (- g'[θc] / t[θc]^Δ[2])
+ruleθ0[g_] := Simplify[g[I θ0]/(-t[I θ0])^Δ[2]/I] - 0.18930
+
+eqLowRHS[F_][m_] := SeriesCoefficient[F[θ], {θ, θc, m}, Assumptions -> Join[{θ < θc, θ > 1}, $Assumptions]]
+
+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
-RFfCoeff[m_] := 0 /; m < 2
-RFfCoeff[m_] := Gamma[m - 1] / π /; m > 1
+eqLow[F_, h_][m_] := eqLowRHS[F][m] - eqLowLHS[h][m]
-ruleB[g_] := π / (2 1.3578383417066) (- g'[θc] / t[θc]^Δ[2])
+eqHighRHS[F_][m_] := D[F[θ], {θ, m} ] /. θ -> 0
-eqLow[n_, h_][m_] := -A B^m RFfCoeff[m] + D[
- RFnonsingular[n][θ] - t[θ]^2 (Gl[h[θ] t[θ]^-Δ[2]] + Log[t[θ]^2]/(8 π)),
- {θ, m} ] / m! /. θ -> θc
+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_] := D[
- RF[n][θ] - (-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]
-eqMid[n_, h_][m_] := D[
- RF[n][θ] - t[θ]^2 Log[h[θ]^2]/(8 Δ[2]π) - h[θ]^((2-α[2])/Δ[2]) Φ[η]
+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
@@ -76,7 +91,10 @@ Gls = {
0.18388370,
-0.6591714,
2.937665,
- -15.61
+ -15.61,
+ 96.76,
+ -679,
+ 5340
}
Ghs = {
@@ -90,7 +108,9 @@ Ghs = {
0,
1457.62,
0,
- -25891
+ -25891,
+ 0,
+ 502000
}
dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)!
@@ -99,9 +119,22 @@ dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)!
GlRules = MapIndexed[dRule[Gl], Gls];
GhRules = MapIndexed[dRule[Gh], Ghs];
-rules[g_] := Join[{B -> ruleB[g]}, ΦRules, GlRules, GhRules]
-
-eq[F_, g_][m_] := Flatten[Join[{eqLow[F, g][#], eqMid[F, g][#]} & /@ Range[0, m], eqHigh[F, g] /@ Range[0, m, 2]]] /. rules[g]
+rules[g_] := Join[ΦRules, GlRules, GhRules]
+(*ξ0 := 0.18930*)
+gC[0] := 1
+tC[0] := 1
+
+eq[F_, g_][m_] := Flatten[Join[{ruleB[g], ruleθ0[g]},{eqLow[F, g][#](*, eqMid[F, g][#]*)} & /@ Range[0, m], eqHigh[F, g] /@ Range[2, m, 2]]] //. rules[g]
+
+newSol[eqs_, oldSol_, newVars_, δ_:0, opts___] := FindRoot[
+ eqs,
+ Join[
+ oldSol /. Rule -> List,
+ Thread[{newVars, δ * RandomVariate[NormalDistribution[], Length[newVars]]}]
+ ],
+ MaxIterations -> 10000,
+ opts
+]
EndPackage[]