论文标题
两次足以容纳危险特征值
Twice is enough for dangerous eigenvalues
论文作者
论文摘要
我们分析了用理性过滤器靶向内部特征值的一类特征值的稳定性。我们表明，即使特征值在滤波器的杆附近，带有有理过滤器的子空间迭代也是可靠的。这些危险的特征值在第一次迭代中造成了很大的圆形错误，但在以后的迭代中是自我校正的。对于具有正交特征向量的矩阵（例如，实时对称或复杂的遗传学），两次迭代足以将圆形误差减少到单位折断的顺序。相比之下，当特征值接近极点时，具有固定杆的理性过滤器加速的Krylov方法通常无法收敛到单位圆形准确性。在Arnoldi的背景下，我们展示了一种简单的重新启动策略，该策略在目标特征台上恢复了完全的精度。
We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.
