ยง Polynomial root finding using QR decomposition

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