§ Polynomial root finding using QR decomposition

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