       Re: Prony method for resonator loss calculations

• To: mathgroup at smc.vnet.net
• Subject: [mg102114] Re: [mg102067] Prony method for resonator loss calculations
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Thu, 30 Jul 2009 05:29:29 -0400 (EDT)
• References: <200907290905.FAA19088@smc.vnet.net>

```After simplifying the code, I get the same results:

Clear[f, v]
ell = 0; c = 1; g = 1; m = 1;
v[x_] = 1;
v[n_][x_] :=
v[n][x] =
I^(ell + 1)*c*
NIntegrate[
y*BesselJ[ell, c*x*y] Exp[-I*(c*g)/2*(x^2 + y^2)] v[n - 1][y], {y,
0, 1}]

Table[v[n], {n, 0, 20}]

{1, 0.293281 + 0.321946 I, -0.0666037 + 0.180302 I, -0.0841789 +
0.0105136 I, -0.0212138 - 0.0308517 I, 0.00778189 - 0.0145778 I,
0.00729276 + 0.0000891848 I,
0.00144596 + 0.00287588 I, -0.000833329 +
0.00115061 I, -0.000620104 - 0.0000929016 I, -0.000089517 -
0.000261862 I, 0.0000844363 - 0.0000882536 I,
0.0000517295 + 0.0000151663 I,
4.56138*10^-6 + 0.0000233507 I, -8.22403*10^-6 +
6.52916*10^-6 I, -4.22856*10^-6 - 1.89685*10^-6 I, -1.15367*10^-7 -
2.04221*10^-6 I, 7.7684*10^-7 - 4.5991*10^-7 I,
3.37972*10^-7 + 2.11023*10^-7 I, -1.40621*10^-8 +
1.7529*10^-7 I, -7.15455*10^-8 + 3.00851*10^-8 I}

Table[v[n], {n, 0, 20}]

{1, 0.122417 + 0.479426 I, -0.163011 + 0.145326 I, -0.0900499 -
0.0343674 I, -0.0048187 - 0.0422661 I, 0.0155841 - 0.0104711 I,
0.00726782 + 0.00398051 I, -0.0000839751 +
0.0036563 I, -0.00144983 + 0.000709563 I, -0.000572462 -
0.000424047 I, 0.0000499212 - 0.000310436 I,
0.000131786 - 0.0000434767 I,
0.0000437982 + 0.0000428136 I, -7.88321*10^-6 +
0.0000258569 I, -0.0000117332 + 2.1615*10^-6 I, -3.22962*10^-6 -
4.15889*10^-6 I, 9.73893*10^-7 - 2.1101*10^-6 I,
1.02464*10^-6 - 4.68554*10^-8 I,
2.26429*10^-7 + 3.92003*10^-7 I, -1.07632*10^-7 +
1.68332*10^-7 I, -8.78196*10^-8 - 7.99464*10^-9 I}

Bobby

On Wed, 29 Jul 2009 04:05:04 -0500, gcarlson <gcarlson at xannah.org> wrote:

> I am trying to compose a Mathematica notebook to implement the Prony
> method as described by Siegman and Miller ("Unstable Optical Resonator
> Loss Calculations Using the Prony Method." Applied Optics 9:2729-2736
> (1970)).
>
> I'm beginning by trying to duplicate the sample calculation in
> Siegman's paper.  I define a functional that I will use to create a
> list of iterated vectors v.
>
> fcnl := Function[{x, f},
>    I^(l+1)*c*NIntegrate[y*BesselJ[l, c*x*y] Exp[-I*(c*g)/2*(x^2 +
> y^2)] f[y] , {y, 0, 1}]]
>
> I define some needed constants.
>
> l := 0
> c := 1
> g := 1
> M := 20
>
> Then I create a module to create the list {Subscript[v,n]}.
>
> Module[{nmax = M},
>   Subscript[v, 0][x_] = 1;
>   Do[Subscript[v, n + 1][x_] = fcnl[x, Subscript[v, n]], {n, 0, nmax}]
>   ]
>
> As a check, I evaluate the 20th iterated vector at the endpoints.
>
> Subscript[v, 20]
> Subscript[v, 20]
>
> -7.15455*10^-8 + 3.00851*10^-8 I
>
> -8.78196*10^-8 - 7.99464*10^-9 I
>
> It appears that the iterated vectors quickly become very small.  Does
> this make sense?
>
> Is this approach doing what I want it to do?
>
> I would appreciate any assistance with or corrections to setting up
> this problem.
>
> Thanks and regards,
>
> Glenn
>
>
>

--
DrMajorBob at bigfoot.com

```

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