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Understanding and modifying classical orthogonal projection methods for large matrix eigenproblems

机译：理解和修改大矩阵特征问题的经典正交投影方法

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Classical orthogonal projection methods for large scale matrix eigen-problems are theoretically analyzed in detail. The methods use Ritz pairs, which are obtained by solving small projected problems, to approximate some eigen-pairs of a given large matrix. The sufficient conditions for the convergence of Ritz pairs are given, showing that in the non-Hermitian case the convergence heavily depends on conditionings of the methods themselves, not only the conditioning of a given problem itself. The convergence of a Ritz value only requires the spetral condition number of the Ritz value itself to be uniformly bounded and does not depend on those of other Ritz values, while that of its corresponding Ritz vector requires the spectral condition numbers of all Ritz values to be uniformly bounded. Since spectral condition numbers of some Ritz values may be arbitrarily large in the non-Hermitian case, Ritz vectors may converge very slowly and even may fail to converge even if corresponding Ritz values do. To overcome such a fatal possible non-convergence of eigenvectors, classical orthogonal projection methods are modified essentially. They keep Ritz values but replaces Ritz vectors by refined approximate eigenvectors. The refined vectors are obtained by solving certain small best approximation problems. A number of results have been established when projection subspaces are a Krylov and a block Krylov subspaces, and some improtant open problems are posed.
机译：从理论上详细分析了大型矩阵特征问题的经典正交投影方法。该方法使用通过解决小型投影问题而获得的Ritz对来近似给定大型矩阵的一些本征对。给出了Ritz对收敛的充分条件，表明在非Hermitian情况下，收敛在很大程度上取决于方法本身的条件，而不仅取决于给定问题本身的条件。 Ritz值的收敛仅要求Ritz值本身的频谱条件数是均匀有界的，而不依赖于其他Ritz值的频谱条件数，而其相应Ritz向量的收敛性则要求所有Ritz值的频谱条件数为统一界。由于在非Hermitian情况下某些Ritz值的频谱条件数可能会任意大，因此即使相应的Ritz值确实收敛，Ritz向量也可能收敛非常慢，甚至可能无法收敛。为了克服特征向量的这种致命的可能的不收敛，对经典的正交投影方法进行了本质上的修改。它们保留Ritz值，但用精细的近似特征向量代替Ritz向量。通过求解某些小的最佳逼近问题来获得精炼的矢量。当投影子空间是Krylov和块Krylov子空间时，已经建立了许多结果，并且提出了一些重要的开放问题。

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