Krylov subspace methods for approximating a matrix function $f(A)$ times avector $v$ are analyzed in this paper. For the Arnoldi approximation to$e^{-\tau A}v$, two reliable a posteriori error estimates are derived from thenew bounds and generalized error expansion we establish. One of them is similarto the residual norm of an approximate solution of the linear system, and theother one is determined critically by the first term of the error expansion ofthe Arnoldi approximation to $e^{-\tau A}v$ due to Saad. We prove that each ofthe two estimates is reliable to measure the true error norm, and the secondone theoretically justifies an empirical claim by Saad. In the paper, byintroducing certain functions $\phi_k(z)$ defined recursively by the givenfunction $f(z)$ for certain nodes, we obtain the error expansion of theKrylov-like approximation for $f(z)$ sufficiently smooth, which generalizesSaad's result on the Arnoldi approximation to $e^{-\tau A}v$. Similarly, it isshown that the first term of the generalized error expansion can be used as areliable a posteriori estimate for the Krylov-like approximation to some othermatrix functions times $v$. Numerical examples are reported to demonstrate theeffectiveness of the a posteriori error estimates for the Krylov-likeapproximations to $e^{-\tau A}v$, $\cos(A)v$ and $\sin(A)v$.
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机译：本文分析了近似矩阵函数$ f（A）$乘向量$ v $的Krylov子空间方法。对于Arnoldi逼近$ e ^ {-\ tau A} v $，我们从新边界和我们建立的广义误差扩展中得出了两个可靠的后验误差估计。它们中的一个类似于线性系统的近似解的残差范数，而另一个则由Arnoldi近似的误差展开的第一项严格地确定为Saad导致的$ e ^ {-\ tau A} v $。我们证明这两个估计中的每一个都可以测量真实误差范数，而第二个估计在理论上证明了Saad的经验主张。在本文中，通过引入由给定函数$ f（z）$为某些节点递归定义的某些函数$ \ phi_k（z）$，我们获得了对于$ f（z）$足够光滑的Krylov型近似的误差展开，这将Saad的结果推广到对$ e ^ {-\ tau A} v $的Arnoldi近似上。类似地，它表明广义误差扩展的第一项可以用作可靠的后验估计，用于对某些其他矩阵函数乘以$ v $的类Krylov近似。数值例子被报道以证明对于$ e ^ {-\ tau A} v $，$ \ cos（A）v $和$ \ sin（A）v $的Krylov-like近似的后验误差估计的有效性。
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