## ยง Polynomial root finding using QR decomposition

- For a polynomial $p(x)$, build the companion matrix $P(x)$.
- Show that the characteristic polynomial $cp(P)$ of the companion matrix $P(x)$ is indeed $p(x)$.
- Find eigenvalues of $P(x)$, which will be roots of $p(x)$, since the eigenvalues of a matrix $M$ are the roots of its characteristic polynomial $cp(M)$.
- We use QR since it is numerically stable. The $Q$ matrix discovered by QR is orthogonal, and hence does not disturb the covariance of the noise on matrix multiplication.