Re: "automatic" piecewise function creation
- To: mathgroup at smc.vnet.net
- Subject: [mg105964] Re: "automatic" piecewise function creation
- From: "Norbert P." <bertapozar at gmail.com>
- Date: Mon, 28 Dec 2009 04:56:52 -0500 (EST)
- References: <hgafri$q8l$1@smc.vnet.net>
On Dec 16, 3:21 am, Giacomo Ciani <giacomo.ci... at gmail.com> wrote:
> Hi all,
>
> I'll be honest, the following is not a real problem to me, as I can
> solve it in an alternative way. However, it is interesting, and a good
> occasion to learn some advanced Mathematica language.
>
> I have to implement a task that is easy to do with "traditional"
> programming techniques (loops). However, I was wondering if there was
> a more compact and elegant way of implementing this exploiting some
> advanced Mathematica function to create lists and tables (or something
> equivalent).
>
> INTRODUCTION TO THE PROBLEM (you can skip this and go to THE PROBLEM,
> but reading it makes everything more comprehensible)
>
> The propagation of a gaussian laser beam through an optical system can
> be conveniently calculated using a formalism in which a complex
> parameter q(z) describes the beam characteristics along the
> propagation direction z, and each optical element is associated with a
> 2x2 matrix A. Each time that a light beam goes through an optical
> element at position z0, its q parameter trasforms as:
>
> q'(z0) = (A_11 q(z0) + A_12)/(A_21 q(z0) + A_22)
>
> Where q(z0) is "immediately before" the optical element, and q'(z0) is
> "immediately after". For your curiosity, this formalism is inherited
> be geometric optics, in which a 2 element vector is used to describe
> the beam, and the matrix formalism seems more natural. Nevertheless,
> using this approach for gaussian beams still has the advantage that
> when you have multiple optical elements in series, instead of
> calculating q at each step you can just multiply the matrices together
> (in the right order!) and the apply the result to q just once.
>
> Then from the parameter q(z) one can calculate, for example, the beam
> width via a specific formula.
>
> Now, I would like to plot the beam width along my optical system. This
> is an example code I use to do that. It describes an optical system
> made of three lenses. The function that I actually plot is the
> propagation of the beam along the direction "d" (my variable), but
> every time it encounters a lens I have to multiply the parameter q
> (propagate untill that point) by the lens matrix, and the start the
> propagation again:
>
> Plot[Piecewise[{
> {w[apl[dst[d], q0]] /. vals, d < lns1Pos},
> {w[apl[dst[d - lns1Pos].lns[-0.050].dst[lns1Pos], q0]] /. vals,
> d <= lns2Pos},
> {w[apl[
> dst[d - lns2Pos].lns[-0.038].dst[
> lns2Pos - lns1Pos].lns[-0.050].dst[lns1Pos], q0]] /. v=
als,
> d <= lns3Pos},
> {w[apl[
> dst[d - lns3Pos].lns[0.2].dst[
> lns3Pos - lns2Pos].lns[-0.038].dst[
> lns2Pos - lns1Pos].lns[-0.050].dst[lns1Pos], q0]] /. v=
als,
> d >= lns3Pos}
> }], {d, 0, 1}]
>
> where: w[q] is a function that returns the width of the beam from its
> parameter q, apl[A,q] is the function that applies the matrix A to the
> parameter q to obtain the new parameter, and lns[f] and dst[l] are
> functions that build the right matrix for lenses of focal length f and
> propagation in space trough a distance l, respectively.
>
> THE PROBLEM:
>
> Summarizing, I have to build a piecewise function with this structure:
>
> Piecewise[{
> {g[f[A,q,x1,z]] z<x2},
> {g[f[B.A,q,x1,x2,z]] z<x3},
> {g[f[C.B.A,q,x1,x2,x3,z]] z<x4},
> ecc...
>
> }]
>
> Where g and f are functions, A,B,C, etc... are matrices (note that
> they are multiplied together, not separated by commas), x1, x2, x3,
> etc... are parameters and z is the free variable.
>
> Doing this starting from a list {{A,x1},{B,x2},{C,x3},...} would be
> rather simple using a loop. But is it possible to do this in
> Mathematica without using a loop?
>
> Hope you will be challenged by the problem!
> (or maybe it's a problem only for me...)
>
> Thanks
>
> Giacomo
Hi Giacomo,
if you don't mind flipping the inequalities, here's a way without any
intermediate expressions:
In[5]:= t={{a,x1},{b,x2},{c,x3},{d,x4},{e,x5}};
Block[{Dot},Piecewise@Reverse@Rest@FoldList[{g[f[Prepend[#1
[[1,1,1]],First@#2],q,Sequence@@Append[#1
[[1,1,3;;-2]],Last@#2],z]],z>=Last@#2}&,{g[f[Dot[],q,z]]},t]]
Out[6]= \[Piecewise]g[f[e.d.c.b.a,q,x1,x2,x3,x4,x5,z]] z>=x5
g[f[d.c.b.a,q,x1,x2,x3,x4,z]] z>=x4
g[f[c.b.a,q,x1,x2,x3,z]] z>=x3
g[f[b.a,q,x1,x2,z]] z>=x2
g[f[a,q,x1,z]] z>=x1
Best,
Norbert