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Given an $n$ by $n$ matrix $A$ and an $n$-vector $b$, along with a rationalfunction $R(z) := D(z )^{-1} N(z)$, we show how to find the optimalapproximation to $R(A) b$ from the Krylov space, $\mbox{span}( b, Ab, \ldots ,A^{k-1} b)$, using the basis vectors produced by the Arnoldi algorithm. To findthis optimal approximation requires running $\max \{ \mbox{deg} (D) ,\mbox{deg} (N) \} - 1$ extra Arnoldi steps and solving a $k + \max \{\mbox{deg} (D) , \mbox{deg} (N) \}$ by $k$ least squares problem. Here {\emoptimal} is taken to mean optimal in the $D(A )^{*} D(A)$-norm. Similar to thecase for linear systems, we show that eigenvalues alone cannot provideinformation about the convergence behavior of this algorithm and we discussother possible error bounds for highly nonnormal matrices.

Optimal Polynomial Approximation to Rational Matrix Functions Using the Arnoldi Algorithm