SIAM Journal on Scientific Computing, Ahead of Print.
We developed a hybrid restarted Lanczos method that combines thick-restarting with Ritz vectors with iteratively refined Ritz vectors to compute a few of the extreme eigenvalues and associated eigenvectors of a large sparse symmetric matrix $A$. The iterative refined Ritz vectors use a scheme, where we replace the approximate eigenvalue in the original refined scheme with the latest computed refined Ritz value until convergence. The thick-restarting schemes have shown to be superior to most other schemes, particularly restarted schemes of linear combinations. However, the simple thick-restarting Lanczos scheme is not available when using refined or iterative refined Ritz vectors. Instead, we use a hybrid restarted scheme that switches between thick-restarted with Ritz vectors and restarting with a judiciously chosen linear combination of iterative refined Ritz vectors. We provide some theoretical results and several computed examples.

中文翻译：

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计算对称矩阵的几个极端特征对的混合迭代精细化方法

《 SIAM科学计算杂志》，预印本。
我们开发了一种混合重新启动的Lanczos方法，该方法结合了Ritz向量的重启动和经过细化的Ritz向量，以计算大型稀疏对称矩阵$ A $的一些极限特征值和相关特征向量。迭代精炼Ritz向量使用一种方案，其中我们用最新计算的精炼Ritz值替换原始精炼方案中的近似特征值，直到收敛为止。厚重启方案已显示出优于大多数其他方案，尤其是线性组合的重启方案。但是，当使用精制或迭代精制Ritz向量时，简单的厚重启Lanczos方案不可用。反而，我们使用混合重新启动方案，该方案在使用Ritz向量的粗重重新启动与通过迭代精炼Ritz向量的明智选择的线性组合重新启动之间进行切换。我们提供了一些理论结果和一些计算示例。