Numerical Convolution

• To: mathgroup at smc.vnet.net
• Subject: [mg116772] Numerical Convolution
• From: "Dan O'Brien" <obrie425 at umn.edu>
• Date: Sun, 27 Feb 2011 04:34:20 -0500 (EST)

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

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