Abstract. By performing an accurate analysis of the convergence, we give a complete theoretical explanation of the experimental behaviour of functional iteration techniques for the computation of the minimal nonnegative solution $G$ of the matrix equation $X=\sum_{i=0}^{+\infty}X^iA_i$, arising in the numerical solution of M/G/1 type Markov chains (here the $A_i$'s are nonnegative $k\times k$ matrices such that the matrix $\sum_{i=0}^{+\infty}A_i$ is column stochastic). Moreover, we introduce a general class of functional iteration methods, which includes the standard methods, and we give an optimality convergence result in this class.

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