Polynomial division without remainder

Question

First, some background on what I'm actually trying to achieve: I have a reflectance filter (a discrete IIR) for use in an FDTD boundary condition:

$$R_{(z)}=\frac{b_0+b_1z^{-1}+b_2z^{-2}}{a_0+a_1z^{-1}+a_2z^{-2}}=\frac{B_{(z)}}{A_{(z)}}$$

As you can see, both the numerator and denominator polynomials are univariate, of the same order, and their powers change by 1 per coefficient. However, I need to turn it into an impedance filter (also represented by an IIR of the same type), according to Kowalczyk 2008 it is done by:

$$\xi_{(z)}=\frac{1+R_{(z)}}{1-R_{(z)}}$$

So my plan was to simply divide $B_{(z)}$ by $A_{(z)}$ to get a single polynomial, and then adjust the first coefficient by $\pm1$.

Only my plan fell at the first hurdle because every polynomial division implementation I've found returns a quotient and remainder. Why are there no implementations that return non-integer (floating point in computer parlance) coefficients? Is it fundamentally impossible for a polynomial division result to be represented in any form other than quotient and remainder?

Or have I just gotten completely confused and the solution to my real problem is far simpler?

Answer 1

Yes it's fundamentally impossible, well at least if you require the quotient to be a polynomial.

Let's look at what (long) division would get you if you divide the polynomials $\phi$ and $\psi$. It's a expression on the form:

$$\phi(x) = q(x)\psi(x) + r(x)$$

this is the general form that the long division algorithm processes. $q$ is the quotient and $r$ is the remainder.

The problem is that we can't generally find a quotient $q$ such that $\phi(x) = q(x)\psi(x)$ (without remainder).

To see an example and why it fails we could assume that $\psi(x)$ is a polynomial of first degree, more concretely let $\psi(x) = x-a$, now we see that it's inevidable that $q(x)\psi(x)$ will have a zero at $x=a$, which means that if we found a quotient without a remainder the polynomial $\psi(x)$ would be required to have a zero for $x=a$ as well - that's not generally true.

Answer 2

What about$$\xi=\frac{1+R}{1-R}=\frac{1+\frac BA}{1-\frac BA}=\frac{A+B}{A-B}$$

So $$\xi = \dfrac{(a_0+b_0)+(a_1+b_1)z^{-1}+(a_2+b_2)z^{-2}} {(a_0-b_0)+(a_1-b_1)z^{-1}+(a_2-b_2)z^{-2}}$$