We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in $O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point operations using $O(\log^2(n/\delta))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-Vargas, Srivastava, 2022).
翻译：我们给出一个基于反向偏移循环的自足随机算法,该算法可以计算任意基质 $M\ in\ mathbb{C\\\n\timen}$,直至以$O(n}4+n ⁇ 3\log§2(n/\delta)\log(n/delta)\log\log\log(n/\delta))$(浮点操作)$,使用$O(\log_2(n/\delta)) 的精确度位数计算。虽然美元(n}4)的复杂度对于大型基质来说是令人望而不可及的,但算法很简单,对于使用受控精度来计算小基质的精度,特别是用于在(Banks, Garza-Vargas, Srivastava, 2022) 的转动QR算法中计算 Ritz值,特别是对于(Banks, Garza-Vargas, Srivastava, 2022)。
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