论文标题
无角集群强大的RITZ值界限的重新启动块eigensolvers
Angle-free cluster robust Ritz value bounds for restarted block eigensolvers
论文作者
论文摘要
解决特征值问题的阻止迭代的收敛速率通常测量近似特征值的Ritz值的误差。 RITZ值的误差通常是根据初始或迭代子空间和与目标特征值关联的不变子空间之间的主角度界定的。因此，由于边界的左侧和右侧使用不同的术语，因此无法根据需要重复使用此类边界。它们必须与其他可能导致高估的界限结合在一起。无角的替代可重复界限仅取决于丽兹值的误差，但在doi中为遗传学特征值问题开创了：10.1515/rnam.1987.2.2.5.371，但仅用于单个极端ritz值。我们将此结果扩展到所有Ritz值，并通过使用非连续特征值来实现聚类特征值的鲁棒性。我们的新界限涵盖了重新开始的块兰开斯方法及其通过移位和放气的修改，并且在数值上是有利的。
Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds that are angle-free and depend only on the errors of the Ritz values have been pioneered for Hermitian eigenvalue problems in doi:10.1515/rnam.1987.2.5.371 but only for a single extreme Ritz value. We extend this result to all Ritz values and achieve robustness for clustered eigenvalues by utilizing nonconsecutive eigenvalues. Our new bounds cover the restarted block Lanczos method and its modifications with shift-and-invert and deflation, and are numerically advantageous.
