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Kappas.m
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BeginPackage["EccentricIMR`Kappas`",
{"EccentricIMR`EccentricPNSymbols`"}];
kappaE;
kappaJ;
$KappaECacheFile;
$KappaJCacheFile;
$KappasFromRepository; (* Set this to False to regenerate the cache;
you also need to set the cache file
variables *)
Begin["`Private`"];
$dataDirectory = FileNameJoin[{FileNameDrop[FindFile["EccentricIMR`Kappas`"],-1],"Data"}];
$kappaERepoCacheFile = FileNameJoin[{$dataDirectory,"kappaE.m"}];
$kappaJRepoCacheFile = FileNameJoin[{$dataDirectory,"kappaJ.m"}];
If[FileExistsQ[$kappaERepoCacheFile] =!= True, Print["ERROR: ", $kappaERepoCacheFile, " not found"]; Abort[]];
If[FileExistsQ[$kappaJRepoCacheFile] =!= True, Print["ERROR: ", $kappaERepoCacheFile, " not found"]; Abort[]];
BesselJprime[p_, x_] := Module[{y, d}, d = D[BesselJ[p, y], y]; d /. y -> x];
(* I have checked that the error made by using the interpolated values
of kappaE or kappaJ from the table instead of the computed values
is of the order of 10^-12 when the table spacing is 0.001. This
was measured by using a spacing ten times smaller and measuring the
largest difference over the range [0,0.4]. *)
computeKappaE[ep_] :=
Module[{p, kappaE, e, term, sumPrev, sum, n, ser},
(* If[ep < 0 || ep >= 1,
Throw["computeKappaE: range error in e"]]; *)
term[e_, p_] := p^3/4(BesselJ[p, p e]^2(1/e^4 - 1/e^2 + 1/3 +
p^2(1/e^4 - 3/e^2 + 3 - e^2)) +
p(-4/e^3 + 7/e - 3e)BesselJ[p, p e] BesselJprime[p, p e] +
BesselJprime[p, p e]^2(1/e^2 - 1 + p^2(1/e^2 - 2 + e^2)));
If[Abs[ep] > 10^-8,
sumPrev = 0;
sum = term[ep, 1];
n = 2;
While[Abs[sum - sumPrev] > 10^-15,
sumPrev = sum;
sum = sum + term[ep, n];
n = n + 1;
(* Print["n = ", n, " sum = ", sum // FullForm];*)];
Return[sum],
(* Print["Using Series"];*)
kappaE = Sum[term[e,p],{p,1,5}];
(* Print["kappaE = ", kappaE];*)
ser = Normal[Series[kappaE,{e,0,4}]];
(* Print[ser];*)
Return[(ser /. e->ep)//N]]];
generateKappaETable[] :=
Table[{e, computeKappaE[e]}, {e, -0.8, 0.8, 0.001}];
saveKappaETable[file_] :=
Export[file, generateKappaETable[], "TSV"];
loadKappaETable[file_] :=
kappaEFn = Interpolation[ReadList[file, Real, RecordLists->True], InterpolationOrder->4];
generateKappaEFn[] :=
Module[{},
Print["Computing kappaE"];
Print[Timing[kappaEFn = Interpolation[generateKappaETable[], InterpolationOrder->4]][[1]]]];
(*generateKappaEFn[];*)
kappaE[e_?NumberQ] :=
(
If[$KappasFromRepository =!= False,
kappaEFn = Get[$kappaERepoCacheFile],
(* else *)
If[StringQ[$KappaECacheFile] && FileExistsQ[$KappaECacheFile],
kappaEFn = Get[$KappaECacheFile],
(* else *)
generateKappaEFn[];
If[StringQ[$KappaECacheFile],
Put[kappaEFn, $KappaECacheFile]]]];
kappaE[ee_?NumberQ] :=
kappaEFn[ee];
kappaE[e]);
computeKappaJ[ep_] :=
Module[{p, kappaJ, e, term, sumPrev, sum, n, ser},
(* If[ep < 0 || ep >= 1,
Throw["computeKappaJ: range error in e"]];*)
term[e_, p_] := p^2/2 Sqrt[
1 - e^2](p(3/e^2 - 2/e^4 - 1)BesselJ[p, p e]^2 + (2/e^3 - 1/e +
2p^2(1/e^3 - 2/e + e))BesselJ[p, p e]BesselJprime[p, p e] +
2p(1 - 1/e^2)BesselJprime[p, p e]^2);
If[Abs[ep] > 10^-8,
sumPrev = 0;
sum = term[ep, 1];
n = 2;
While[Abs[sum - sumPrev] > 10^-15,
sumPrev = sum;
sum = sum + term[ep, n];
n = n + 1;
(* Print["n = ", n, " sum = ", sum // FullForm];*)];
Return[sum],
(* Print["Using Series"];*)
kappaJ = Sum[term[e,p],{p,1,5}];
(* Print["kappaJ = ", kappaJ];*)
ser = Normal[Series[kappaJ,{e,0,4}]];
Return[(ser /. e->ep)//N]]];
generateKappaJTable[] :=
Table[{e, computeKappaJ[e]}, {e, -0.8, 0.8, 0.001}];
generateKappaJFn[] :=
Module[{},
Print["Computing kappaJ"];
Print[Timing[kappaJFn = Interpolation[generateKappaJTable[], InterpolationOrder->4]][[1]]]];
(*generateKappaJFn[];*)
kappaJ[e_?NumberQ] :=
(
(* TODO: eliminate duplication with kappaE above *)
If[$KappasFromRepository =!= False,
kappaJFn = Get[$kappaJRepoCacheFile],
(* else *)
If[StringQ[$KappaJCacheFile] && FileExistsQ[$KappaJCacheFile],
kappaJFn = Get[$KappaJCacheFile],
(* else *)
generateKappaJFn[];
If[StringQ[$KappaJCacheFile],
Put[kappaJFn, $KappaJCacheFile]]]];
kappaJ[ee_?NumberQ] :=
kappaJFn[ee];
kappaJ[e]);
(* generateKappaEFn[]; *)
End[];
EndPackage[];