Re: Numerical Convolution
- To: mathgroup at smc.vnet.net
- Subject: [mg116788] Re: Numerical Convolution
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Mon, 28 Feb 2011 05:00:05 -0500 (EST)
----- Original Message ----- > From: "Dan O'Brien" <obrie425 at umn.edu> > To: mathgroup at smc.vnet.net > Sent: Sunday, February 27, 2011 3:34:20 AM > Subject: [mg116772] Numerical Convolution > I'm attempting to write a function that can be used as the expression > in > NonlinearModelFit. In general, the function is a convolution of a > gaussian and any arbitrary function. Here I have defined it using the > complex functions \[Chi]R > > \[Chi]R[f_, a_, \[CapitalGamma]_, \[Omega]v_, \[Omega]_] := > f (Sqrt[Abs[a]] Abs[\[CapitalGamma]])/(-(\[Omega] - \[Omega]v) - > I Abs[\[CapitalGamma]]) > > F[f1_, A1_, \[CapitalGamma]1_, \[Omega]v1_, f2_, > A2_, \[CapitalGamma]2_, \[Omega]v2_, \[Omega]_] := > Abs[\[Chi]R[f1, > A1, \[CapitalGamma]1, \[Omega]v1, \[Omega]] + \[Chi]R[f2, > A2, \[CapitalGamma]2, \[Omega]v2, \[Omega]]]^2 > > (*Cannot evaluate this integral*) > conv[f1_, A1_, \[CapitalGamma]1_, \[Omega]v1_, f2_, > A2_, \[CapitalGamma]2_, \[Omega]v2_, \[Omega]_] := > Convolve[PDF[NormalDistribution[0, 3], x], > F[f1, A1, \[CapitalGamma]1, \[Omega]v1, f2, > A2, \[CapitalGamma]2, \[Omega]v2, x], x, \[Omega]] > > conv[1, 20, 5, 1702, -1, 10, 3, 1692, \[Omega]](*I stop it after a \ > few minutes on my machine,windows xp sp3 Mathematica 8.0*) > > (*So my thought was to try to set up a numerical convolution in the \ > vicinity of my domain of interest.*) > > Nconv[f1_, A1_, \[CapitalGamma]1_, \[Omega]v1_, f2_, > A2_, \[CapitalGamma]2_, \[Omega]v2_] := > Interpolation[ > ParallelTable[{i, > NIntegrate[ > PDF[NormalDistribution[0, 3], x] F[f1, > A1, \[CapitalGamma]1, \[Omega]v1, f2, > A2, \[CapitalGamma]2, \[Omega]v2, > i - x], {x, -\[Infinity], \[Infinity]}]}, {i, 1620, 1740}]] > > (*So the interpolation function would be used to fit my data in \ > NonlinearModelFit.But this doesn't work to use as the model \ > expression and I'm not sure where to begin to make it work.Even to \ > plot it,it must be evaluated first and then used*) > > g = Nconv[1, 20, 5, 1702, -1, 10, 3, 1692] > > Plot[{g[\[Omega]], > F[1, 20, 5, 1702, -1, 10, 3, 1692, \[Omega]]}, {\[Omega], 1620, > 1740}, PlotRange -> All] > > Is there a better way to do this? Any help is very much appreciated. > > -Dan This might be along the lines of what you want. conv[f1_, A1_, \[CapitalGamma]1_, \[Omega]v1_, f2_, A2_, \[CapitalGamma]2_, \[Omega]v2_, \[Omega]_?NumericQ] := Re[NIntegrate[ PDF[NormalDistribution[0, 3], x]* F[f1, A1, \[CapitalGamma]1, \[Omega]v1, f2, A2, \[CapitalGamma]2, \[Omega]v2, \[Omega] - x], {x, -Infinity, Infinity}]] Nconv[f1_, A1_, \[CapitalGamma]1_, \[Omega]v1_, f2_, A2_, \[CapitalGamma]2_, \[Omega]v2_] := Interpolation[ Table[conv[f1, A1, \[CapitalGamma]1, \[Omega]v1, f2, A2, \[CapitalGamma]2, \[Omega]v2, i], {i, 1620, 1740}], {i, 1620, 1740}] In[43]:= Timing[g = Nconv[1, 20, 5, 1702, -1, 10, 3, 1692];] Out[43]= {3.042000000000002, Null} Daniel Lichtblau Wolfram Research