       Re: Numerical Convolution

• To: mathgroup at smc.vnet.net
• Subject: [mg116790] Re: Numerical Convolution
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Mon, 28 Feb 2011 05:00:26 -0500 (EST)

```These are the same:

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;
positive =
FullSimplify[
ComplexExpand[
F[f1, A1, \[CapitalGamma]1, \[Omega]v1, f2,
A2, \[CapitalGamma]2, \[Omega]v2, \[Omega]],
TargetFunctions -> {Re, Im}],
f1 > 0 && A1 > 0 && \[CapitalGamma]1 > 0 && \[Omega]v1 > 0 &&
f2 < 0 &&
A2 > 0 && \[CapitalGamma]2 > 0 && \[Omega]v2 > 0 && \[Omega] >
0];
negative =
FullSimplify[
ComplexExpand[
F[f1, A1, \[CapitalGamma]1, \[Omega]v1, f2,
A2, \[CapitalGamma]2, \[Omega]v2, \[Omega]],
TargetFunctions -> {Re, Im}],
f1 > 0 && A1 > 0 && \[CapitalGamma]1 > 0 && \[Omega]v1 > 0 &&
f2 < 0 &&
A2 > 0 && \[CapitalGamma]2 > 0 && \[Omega]v2 > 0 && \[Omega] <
0];
positive === negative

True

so we get an equivalent function (near the parameters from your example)
WITHOUT imaginaries, with

eff[f1_, A1_, \[CapitalGamma]1_, \[Omega]v1_, f2_,
A2_, \[CapitalGamma]2_, \[Omega]v2_, \[Omega]_] = positive

(f2 \[CapitalGamma]2 (A2 f2 \[CapitalGamma]2 (\[CapitalGamma]1^2 + (\
\[Omega] - \[Omega]v1)^2) +
2 Sqrt[A1 A2]
f1 \[CapitalGamma]1 (\[CapitalGamma]1 \[CapitalGamma]2 + (\
\[Omega] - \[Omega]v1) (\[Omega] - \[Omega]v2))) +
A1 f1^2 \[CapitalGamma]1^2 (\[CapitalGamma]2^2 + (\[Omega] - \
\[Omega]v2)^2))/((\[CapitalGamma]1^2 + (\[Omega] - \[Omega]v1)^2) (\
\[CapitalGamma]2^2 + (\[Omega] - \[Omega]v2)^2))

I also have have an approximation:

NIntegrate[
eff[1, 20, 5, 1702, -1, 10, 3, 1692, x] PDF[NormalDistribution[0, 3],
y - x], {x, 1650, 1800}, {y, 1701 - 35, 1701 + 35}]

269.03

The x limits were based on a look at the graph:

Plot[eff[1, 20, 5, 1702, -1, 10, 3, 1692, x], {x, 1692 - 10^2,
1692 + 10^2}, PlotRange -> All]

and locating the maximum:

NMaximize[{eff[1, 20, 5, 1702, -1, 10, 3, 1692, x],
1650 < x < 1750}, x]

{19.1435, {x -> 1700.97}}

That's at most a start to the general problem, of course.

Bobby

On Sun, 27 Feb 2011 03:34:20 -0600, Dan O'Brien <obrie425 at umn.edu> wrote:

> 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
>
>

--
DrMajorBob at yahoo.com

```

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