Although I have a PhD in Electrical Engineering I am not a mathematician and here is a problem I cannot solve on my own. I would really appreciate your help.
For calculating dedicated audio filters my goal is to transform a very long polynomial (64k coefficients) in a certain way (see below) without factorizing it. There are certain transformations well known (please see here https://brilliant.org/wiki/transforming-roots-of-polynomial/) but all of them did not cover my problem.
Let us assume that the existing polynomial has real and conjugate complex roots $r_n$. The roots can be found inside and outside the unit circle but none of them is on the unit circle so $\lvert r_n \rvert \neq 1$.
I want to scale the roots $r_n$ by a factor $f \leq 1$ shifting them towards the center of the unit circle. This means that $r_n’= f\cdot r_n$. Now the challenge is that $f$ is not constant but a function of $\arg(r_n)$ so $r_n’ = f(\arg(r_n)) \cdot r_n$.
$f(\arg(r_n))$ is decreasing in $\arg(r_n)$. This means for $\arg(r_n)=0$ $f=1$ and between $0<\arg(r_n)\leq \pi/2$, $f$ becomes smaller the bigger $\arg(r_n)$ gets.
This means roots with a small $\arg(r)$ are shifted slightly towards zero, roots with a bigger $\arg(r)$ will be shifted stronger. Any idea how this can be achieved without factorizing? Thanks a lot Best Charly