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We investigate the problem of approximating the matrix function $f(A)$ by$r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound forthe relative interpolation error $1-r/f$ on the spectral interval of $A$. Byminimizing this upper bound over all interpolation points, we obtain a new,simple and sharp a priori bound for the relative interpolation error. We thenconsider three different approaches of representing and computing the rationalinterpolant $r$. Theoretical and numerical evidence is given that any of thesemethods for a scalar argument allows to achieve high precision, even in thepresence of finite precision arithmetic. We finally investigate the problem ofefficiently evaluating $r(A)$, where it turns out that the relative error for amatrix argument is only small if we use a partial fraction decomposition for$r$ following Antoulas and Mayo. An important role is played by a new stoppingcriterion which ensures to automatically find the degree of $r$ leading to asmall error, even in presence of finite precision arithmetic.

On the rational approximation of Markov functions,with applications to the computation of Markovfunctions of Toeplitz matrices