# Linear/Rational compositions

1. Feb 27, 2006

### shaner-baner

"Linear/Rational" compositions

I wanted to share an application of linear algebra that I recently used at work. I hope you guys enjoy it to.
This problem arose in connection with a recursion of the form
$$S^k=f(S^{k-1} )$$. where $$f=\frac{\ a+bx}{c+dx}$$
a little thought shows that the space of rational functions that are quotients of linear functions is 'like' M(2x2) (ignoring for a moment that the matrix representation is not unique).
$$\left(\begin{array}{cc} a&b\\c&d\end{array}\right)$$

It's even easy to define a product on this space. It represents the composition of two of these functions which is itself a rational function with degree 1 top and bottom. So the space is closed under this operation. this means that the quantity I was interested in could be expressed as a product of these matrices (not the standard product) so
$$S^k=(C^kC^{k-1}C^{k-2}....C^1)S^0$$. Lucky for me, these compositions are associative, and from calculation to calculation only C(3),C(2),and
C(1) changed. So I just calculated the 'product' of C(k)...C(4) once and stored it for later use, saving me bundles of time.
Also of interest to me was that in the system I was modeling, S(k) expressed recursively, so it depended on all the S's before it, but the C's each represent seperate part of my system, which is of practical interest.
Just goes to show, a little mathematics goes along way.

Last edited: Feb 27, 2006