Convergence of pure and relaxed Newton methods for solving a matrix polynomial equation arising in stochastic models

We consider a matrix polynomial equation which has the form of AnXn+An−1Xn−1+⋯+A0=0 where An,An−1,…,A0 and X are square matrices assuming the positivity of coefficients from stochastic models. The monotone convergence of Newtonʼs method for solving the equation is considered and we show that the elementwise minimal nonnegative solution can be found by the method with the zero starting matrix. Moreover, the relaxed Newton method preserving the monotonicity result is provided. Finally, numerical experiments show that our method reduces the number of iterations significantly.