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-rw-r--r--new_schofield.nb56032
-rw-r--r--schofield.wl172
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 @@
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AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->{True, True},
AxesLabel->{None, None},
- AxesOrigin->{0., 1.18113318389246},
+ AxesOrigin->{0, 0},
DisplayFunction->Identity,
Frame->{{False, False}, {False, False}},
FrameLabel->{{None, None}, {None, None}},
@@ -148915,18 +154115,27 @@ Cell[BoxData[
GridLines->{None, None},
GridLinesStyle->Directive[
GrayLevel[0.5, 0.4]],
+ ImagePadding->All,
Method->{
- "OptimizePlotMarkers" -> True, "OptimizePlotMarkers" -> True,
+ "DefaultBoundaryStyle" -> Automatic,
+ "DefaultGraphicsInteraction" -> {
+ "Version" -> 1.2, "TrackMousePosition" -> {True, False},
+ "Effects" -> {
+ "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2},
+ "Droplines" -> {
+ "freeformCursorMode" -> True,
+ "placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" ->
+ AbsolutePointSize[6], "ScalingFunctions" -> None,
"CoordinatesToolOptions" -> {"DisplayFunction" -> ({
- Identity[
+ (Identity[#]& )[
Part[#, 1]],
- Identity[
+ (Identity[#]& )[
Part[#, 2]]}& ), "CopiedValueFunction" -> ({
- Identity[
+ (Identity[#]& )[
Part[#, 1]],
- Identity[
+ (Identity[#]& )[
Part[#, 2]]}& )}},
- PlotRange->{{0., 9.}, {1.18113318389246, 1.196372897646665}},
+ PlotRange->{{0, 1.1}, {-0.29799297961973337`, 0.5283576437115491}},
PlotRangeClipping->True,
PlotRangePadding->{{
Scaled[0.02],
@@ -148934,583 +154143,149 @@ Cell[BoxData[
Scaled[0.05],
Scaled[0.05]}},
Ticks->{Automatic, Automatic}]], "Output",
- CellChangeTimes->{3.8395814031089983`*^9},
- CellLabel->
- "Out[864]=",ExpressionUUID->"fa7fd377-360b-4502-a977-985fa8362db0"]
+ CellChangeTimes->{{3.840877493560769*^9, 3.840877534824695*^9},
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-Cell[14880196, 284104, 1529, 43, 24, "Input",ExpressionUUID->"bdb2aaf8-6c60-4fd6-b0aa-d74503ebfe5d"],
-Cell[14881728, 284149, 216, 4, 39, "Output",ExpressionUUID->"672f3cb6-b559-424b-b6e9-dff18bdab931"]
+Cell[13104121, 248760, 1529, 43, 24, "Input",ExpressionUUID->"bdb2aaf8-6c60-4fd6-b0aa-d74503ebfe5d"],
+Cell[13105653, 248805, 216, 4, 39, "Output",ExpressionUUID->"672f3cb6-b559-424b-b6e9-dff18bdab931"]
}, Open ]],
Cell[CellGroupData[{
-Cell[14881981, 284158, 2328, 62, 24, "Input",ExpressionUUID->"9581e610-c27e-4ccd-9de0-7344556f685a"],
-Cell[14884312, 284222, 584, 12, 22, "Message",ExpressionUUID->"926f5636-f494-4b80-9e0f-998a05d07881"],
-Cell[14884899, 284236, 420, 8, 25, "Output",ExpressionUUID->"3c60fc73-ad7e-4b33-a3eb-fdc35776a150"]
+Cell[13105906, 248814, 2328, 62, 24, "Input",ExpressionUUID->"9581e610-c27e-4ccd-9de0-7344556f685a"],
+Cell[13108237, 248878, 584, 12, 22, "Message",ExpressionUUID->"926f5636-f494-4b80-9e0f-998a05d07881"],
+Cell[13108824, 248892, 420, 8, 25, "Output",ExpressionUUID->"3c60fc73-ad7e-4b33-a3eb-fdc35776a150"]
}, Open ]],
-Cell[14885334, 284247, 1234, 32, 24, "Input",ExpressionUUID->"3a793a2c-1a2c-433a-873f-55e7861acacf"],
+Cell[13109259, 248903, 1234, 32, 24, "Input",ExpressionUUID->"3a793a2c-1a2c-433a-873f-55e7861acacf"],
Cell[CellGroupData[{
-Cell[14886593, 284283, 379, 9, 24, "Input",ExpressionUUID->"6a4616ab-41e4-44cc-8806-ee9851e65934"],
-Cell[14886975, 284294, 21750, 374, 177, "Output",ExpressionUUID->"7ae541cc-49f0-4346-9e8f-3995505a8e9f"]
+Cell[13110518, 248939, 379, 9, 24, "Input",ExpressionUUID->"6a4616ab-41e4-44cc-8806-ee9851e65934"],
+Cell[13110900, 248950, 21750, 374, 177, "Output",ExpressionUUID->"7ae541cc-49f0-4346-9e8f-3995505a8e9f"]
}, Open ]],
-Cell[14908740, 284671, 2535, 66, 24, "Input",ExpressionUUID->"02eee1a8-8fdf-4ef6-8fa3-737b3067106e"],
+Cell[13132665, 249327, 2535, 66, 24, "Input",ExpressionUUID->"02eee1a8-8fdf-4ef6-8fa3-737b3067106e"],
Cell[CellGroupData[{
-Cell[14911300, 284741, 491, 10, 24, "Input",ExpressionUUID->"8858ddfc-1e71-4778-a0b8-512abba3d901"],
-Cell[14911794, 284753, 22737, 390, 179, "Output",ExpressionUUID->"3a7e88f8-603a-4c66-8a1b-065c43aff3cd"]
+Cell[13135225, 249397, 491, 10, 24, "Input",ExpressionUUID->"8858ddfc-1e71-4778-a0b8-512abba3d901"],
+Cell[13135719, 249409, 22737, 390, 179, "Output",ExpressionUUID->"3a7e88f8-603a-4c66-8a1b-065c43aff3cd"]
}, Open ]],
-Cell[14934546, 285146, 3849, 95, 24, "Input",ExpressionUUID->"954f68de-17bb-4b28-b089-18f4c1840d53"],
+Cell[13158471, 249802, 3849, 95, 24, "Input",ExpressionUUID->"954f68de-17bb-4b28-b089-18f4c1840d53"],
Cell[CellGroupData[{
-Cell[14938420, 285245, 448, 10, 24, "Input",ExpressionUUID->"fd8d9fbd-6567-43cc-a4dc-4c986b02e466"],
-Cell[14938871, 285257, 26854, 485, 178, "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[]