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In order to determine the stationary distribution for discrete timequasi-birth-death Markov chains, it is necessary to find the minimalnonnegative solution of a quadratic matrix equation. We apply theNewton-Shamanskii method for solving the equation. We show that the sequence ofmatrices generated by the Newton-Shamanskii method is monotonically increasingand converges to the minimal nonnegative solution of the equation. Numericalexperiments show the effectiveness of our method.

The Newton-Shamanskii method for solving a quadratic matrix equation arising in quasi-birth-death problems