MATH319 Slides 149 Example of proportional-integral feedback 151 Euclidean algorithm

150 Highest common factor and common zeros

Let $G$ be a complex rational function. Then we can write $G=P/Q$ for complex polynomials $P$ and $Q$ . Note that the Euclidean algorithm can be carried out in exact arithmetic–no need to find roots of polynomial equations.

Lemma

Nonzero complex polynomials $P$ and $Q$ have no common complex zeros if and only if there exist complex polynomials $M$ and $N$ such that

PM+QN=1.

MATLAB can find the greatest common divisor (highest common factor) for polynomials via command

$>>$ [g,M,N]=gcd(P,Q)