Let R(z) be a matrix function. We propose modified Newton's method to calculate zero points of detR(z). By the modified method, we can obtain accurate zero points by simple iterations. We also extend this problem to a multivariable case. Applications to the spectral analysis of M/G/1 type Markov chains are discussed. Important characteristics of these chains, e.g., the boundary vector and the matrix G, can be derived from zero points of a matrix function and corresponding null vectors. Numerical results are shown.
