diff options
| -rw-r--r-- | new_schofield.nb | 56032 | ||||
| -rw-r--r-- | schofield.wl | 172 |
2 files changed, 10544 insertions, 45660 deletions
diff --git a/new_schofield.nb b/new_schofield.nb index 554243e..d2f64fb 100644 --- a/new_schofield.nb +++ b/new_schofield.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 15107921, 287965] -NotebookOptionsPosition[ 14965931, 285751] -NotebookOutlinePosition[ 14966330, 285767] -CellTagsIndexPosition[ 14966287, 285764] +NotebookDataLength[ 13344452, 252763] +NotebookOptionsPosition[ 13189856, 250407] +NotebookOutlinePosition[ 13190255, 250423] +CellTagsIndexPosition[ 13190212, 250420] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -30,9 +30,9 @@ Cell[BoxData[ 3.827916450999913*^9}, {3.839304619292111*^9, 3.839304619735818*^9}, { 3.839409661432328*^9, 3.839409662064075*^9}, {3.839479110706854*^9, 3.839479110850239*^9}, {3.839569665835891*^9, 3.839569666219857*^9}, { - 3.83957102205329*^9, 3.8395710222207737`*^9}}, - CellLabel-> - "In[1560]:=",ExpressionUUID->"e595a667-7f25-4f98-9f12-a321a236d113"], 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"Output",ExpressionUUID->"b7f39439-0c1d-40f8-af40-5fed6fd931de"], -Cell[14965728, 285744, 175, 3, 25, "Output",ExpressionUUID->"2b7f8ec2-0126-4787-92e3-a6d9dd600de2"] +Cell[13162345, 249901, 448, 10, 24, "Input",ExpressionUUID->"fd8d9fbd-6567-43cc-a4dc-4c986b02e466"], +Cell[13162796, 249913, 26854, 485, 178, "Output",ExpressionUUID->"b7f39439-0c1d-40f8-af40-5fed6fd931de"], +Cell[13189653, 250400, 175, 3, 25, "Output",ExpressionUUID->"2b7f8ec2-0126-4787-92e3-a6d9dd600de2"] }, Open ]] }, Closed]] } diff --git a/schofield.wl b/schofield.wl index 83cf6ac..1c750a9 100644 --- a/schofield.wl +++ b/schofield.wl @@ -1,7 +1,7 @@ BeginPackage["Schofield`"] -$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 0} +$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0} β[D_:2] := Piecewise[ { @@ -32,10 +32,10 @@ $Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 0} Δ[D_:2] := β[D] δ[D] -OverBar[s] := 1.357838341706595496 +OverBar[s] := 2^(1/12) Exp[-1/8] Glaisher^(3/2) -t[θ_] := ((θ)^2 - 1) -h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]LegendreP[(2 * i + 1), θ/θc], {i, 0, n}] +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/( @@ -45,31 +45,63 @@ 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) +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] LegendreP[(2 i), θ/θc] , {i, 0, n}] +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] LegendreP[(2 i), θ/θc], {i, 0, n}] + 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] -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 - 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 + 1.37 * (g[I θ0]/ I)^(2 / Δ[2]) * (-η[g]'[I θ0] / (2 θ0 I))^(5/6) +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] @@ -107,16 +139,16 @@ eqMid[F_, h_][m_] := D[ } Gls = { - Around[0, δ0], + 0, -OverBar[s], - −0.0489532897203, - 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, + −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, @@ -125,17 +157,17 @@ Gls = { } Ghs = { - Around[0, δ0], - Around[0, δ0], - -1.845228078232838, - 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, + -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, @@ -148,19 +180,23 @@ dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)! 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*) +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 -eq[n_, g_][m_, p_, q_] := Flatten[Join[{ruleB[g], ruleθ0[g], g'[0] - 1, ruleAH[g], ruleAL[g]}, eqLow[n, g][#] & /@ Range[0, m],eqMid[RF[n], g][#] & /@ Range[0, p], eqHigh[n, g] /@ Range[0, q, 2]]] //. rules /. 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] - (* *) -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 +formResiduals[data_, functions_, δ_:0] := 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, @@ -172,5 +208,55 @@ newSol[eqs_, oldSol_, newVars_, δ_:0, γ_:0, opts___] := FindRoot[ opts ] +levenburgMarquardt[r_, \[Beta]0_, \[Lambda]0_ : 1, \[Nu]_ : 2] := + Module[ + { + n, \[Beta], new\[Beta], rc, J, I, oldJ, newJ, oldr, newr, M, + g, \[Delta], newcost, \[Lambda], oldcost, temp + }, + n = Length[\[Beta]0]; \[Lambda] = \[Lambda]0; \[Beta] = \[Beta]0; + PrintTemporary["Compiling the Jacobian..."]; + rc = Compile[{{x, _Real, 1}}, + Evaluate[ + r /. Thread[Rule[First /@ \[Beta], Part[x, #] & /@ Range[n]]]]]; + J = Compile[{{x, _Real, 1}}, + Evaluate[ + D[r, {First /@ \[Beta]}] /. + Thread[Rule[First /@ \[Beta], Part[x, #] & /@ Range[n]]]]]; + PrintTemporary["Beginning the algorithm."]; + oldJ = J[\[Beta][[All, 2]]]; + oldr = rc[\[Beta][[All, 2]]]; + oldcost = Re[Total[oldr^2]]; + new\[Beta] = \[Beta]; + g = Re[Transpose[oldJ] . oldr]; + M = Re[Transpose[oldJ] . oldJ]; + \[Delta] = LinearSolve[M + \[Lambda] DiagonalMatrix[Diagonal[M]], g]; + PrintTemporary[Dynamic[oldcost]] + While[Norm[\[Delta]] > 10^-15, + \[Delta] = + LinearSolve[M + \[Lambda]/\[Nu] DiagonalMatrix[Diagonal[M]], g]; + new\[Beta][[All, 2]] -= \[Delta]; + newr = rc[new\[Beta][[All, 2]]]; + newcost = Re[Total[newr^2]]; + While[newcost > oldcost, + \[Delta] = + LinearSolve[M + \[Lambda] DiagonalMatrix[Diagonal[M]], g]; + new\[Beta] = \[Beta]; + new\[Beta][[All, 2]] -= \[Delta]; + newr = rc[new\[Beta][[All, 2]]]; + newcost = Re[Total[newr^2]]; + \[Lambda] *= \[Nu]; + ]; + \[Lambda] /= \[Nu]; + oldcost = newcost; + oldr = newr; + \[Beta] = new\[Beta]; + oldJ = J[\[Beta][[All, 2]]]; + g = Re[Transpose[oldJ] . oldr]; + M = Re[Transpose[oldJ] . oldJ]; + ]; + {newcost, \[Beta]} + ] + EndPackage[] |