"automatic" piecewise function creation
- To: mathgroup at smc.vnet.net
- Subject: [mg105711] "automatic" piecewise function creation
- From: Giacomo Ciani <giacomo.ciani at gmail.com>
- Date: Wed, 16 Dec 2009 06:18:48 -0500 (EST)
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]] /. vals,
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]] /. vals,
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