We perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm. We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and take the non-orthonormality of the computed Krylov basis into account. We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed matrix, and we give bounds for the perturbation. We show that the shift-and-invert Arnoldi algorithm is backward stable if the condition number of the small Hessenberg matrix is not too large. This condition is then relaxed using implicit restarts. Moreover, we give notes on the Hermitian case, considering Hermitian backward errors, and finally, we use our analysis to derive a sensible breakdown condition.

中文翻译：

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shift-and-invert Arnoldi算法的后向误差分析

我们对不精确的移位和反转 Arnoldi 算法进行后向误差分析。我们考虑了产生的线性系统的解以及正交化步骤中的不精确性，并考虑了计算出的 Krylov 基的非正交性。我们证明计算基和 Hessenberg 矩阵满足扰动矩阵的精确移位和逆 Krylov 关系，并且我们给出了扰动的界限。我们表明，如果小 Hessenberg 矩阵的条件数不太大，则移位和反转 Arnoldi 算法是向后稳定的。然后使用隐式重新启动放宽此条件。此外，我们对 Hermitian 情况给出了注释，考虑了 Hermitian 向后误差，最后，我们使用我们的分析来推导出合理的击穿条件。