Optimal Polynomial Approximation to Rational Matrix Functions Using the Arnoldi Algorithm

Given an $n$ by $n$ matrix $A$ and an $n$-vector $b$, along with a rational function $R(z) := D(z )^{-1} N(z)$, we show how to find the optimal approximation 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 find this 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 {\em optimal} is taken to mean optimal in the $D(A )^{*} D(A)$-norm. Similar to the case for linear systems, we show that eigenvalues alone cannot provide information about the convergence behavior of this algorithm and we discuss other possible error bounds for highly nonnormal matrices.