Given a nonsymmetric matrix $A\in {ℝ}^{n\times n}$ and two unit norm vectors, the two‐sided Krylov subspace methods construct a pair of bases for two Krylov subspaces with respect to A and A^T, respectively. In practical calculations, however, the two subspaces spanned by the computed bases may not be Krylov subspaces. Given two subspaces $𝒦$ and $ℒ$, in [G. Wu et al, Toward backward perturbation bounds for approximate dual Krylov subspaces, BIT, 53 (2013), pp. 225‐239], the authors considered how to determine a backward perturbation E whose norm is as small as possible, such that $𝒦$ and $ℒ$ are Krylov subspaces of A + E and (A + E)^T, respectively. However, as the two bases used are biorthonormal, their results are nonoptimal in terms of unitarily invariant norms, and the perturbation bound can be greatly overestimated. In this work, we revisit this problem and use orthonormal bases instead of biorthonormal bases to derive new perturbation bounds. We propose two new strategies, the first one focuses on choosing optimal orthonormal basis matrices, and the second one resorts to solving small‐sized generalized Sylvester matrix equations. Numerical experiments show that our bounds improve the existing one substantially.

中文翻译：

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确定近似两侧Krylov子空间的后向摄动界的新策略

给定一个非对称矩阵 $\mathrm{一种}\in {ℝ}^{ñ\times ñ}$和两个单位范式向量，双向Krylov子空间方法针对两个Krylov子空间分别针对A和A ^T构造了一对底。但是，在实际计算中，被计算基数跨越的两个子空间可能不是Krylov子空间。给定两个子空间$𝒦$ 和 $ℒ$[ G. Wu等，近似对偶Krylov子空间的向后摄动界，BIT，53（2013），第225-239页]中，作者考虑了如何确定范数尽可能小的后向摄动E，这样$𝒦$ 和 $ℒ$分别是A  +  E和（A  +  E）^T的Krylov子空间。但是，由于使用的两个碱基是双正交的，因此它们的结果在in不变范数方面不是最佳的，并且扰动范围可能会大大高估。在这项工作中，我们重新审视了这个问题，并使用正交正态基础而不是双正交正态基础来推导新的摄动界。我们提出了两种新策略，第一种策略侧重于选择最优正交基矩阵，第二种策略求助于求解小型广义Sylvester矩阵方程。数值实验表明，我们的边界大大改善了现有边界。