Durand-Kerner method for the real roots

Given a univariate real polynomial, this paper considers calculating all the real zero-points of the polynomial simultaneously. Among several numerical methods for calculating zeropoints of a univariate polynomial, Durand-Kerner method is quite useful because it is most stable to converge to the zero-points. On the basis of Durand-Kerner method, we propose two methods for calculating the real zero-points of a univariate polynomial simultaneously. Convergence and error analysis of our methods are discussed. We compared our methods, the original Durand-Kerner method and Newton's method and found that 1) our methods are more stable than Newton's method but less than the original Durand-Kerner method, and 2) they are more efficient than the original Durand-Kerner method but less than Newton's method. We conclude that our methods are useful when good initial values of the zero-points are known.