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.

翻译：给定一个n×n矩阵A和一个n维向量b，以及有理函数R(z) := D(z)^{-1} N(z)，我们展示了如何利用Arnoldi算法生成的基向量，从Krylov空间span(b, Ab, …, A^{k-1}b)中找到对R(A)b的最优逼近。要找到这个最优逼近，需要运行max{deg(D), deg(N)} - 1次额外的Arnoldi步骤，并求解一个k + max{deg(D), deg(N)}行k列的最小二乘问题。这里的"最优"是指在D(A)^* D(A)范数下的最优。与线性系统的情况类似，我们证明了特征值本身无法提供该算法收敛行为的信息，并讨论了针对高度非正规矩阵的其他可能的误差界。