非精确Rayleigh商迭代与MINRES的收敛性
On Convergence of the Inexact Rayleigh Quotient Iteration with MINRES
AI总结：
针对Hermitian非精确Rayleigh商迭代，提出基于均匀正定条件的新收敛理论，证明与MINRES结合时可在更宽松的内迭代容差下实现三次、二次和线性收敛。
摘要：
对于Hermitian非精确Rayleigh商迭代（RQI），我们提出了一种新的通用理论，该理论独立于移位内线性系统的迭代求解器。该理论表明，在称为均匀正定条件的新条件下，该方法至少二次收敛，该条件允许在外迭代$k$时内容差$ξ_k\geq 1$，并且可能比文献中常用的条件$ξ_k\leqξ<1$（其中$ξ$为不接近1的常数）弱得多。我们考虑了使用无预条件和调谐预条件MINRES方法求解线性系统的非精确RQI的收敛性。推导了MINRES所得残差的一些吸引人的性质。基于这些性质和新的一般理论，我们进行了更精细的分析，并建立了一系列新的收敛结果。设$\|r_k\|$为外迭代$k$时近似特征对的残差范数。那么所有现有的三次和二次收敛结果分别要求$ξ_k=O(\|r_k\|)$和$ξ_k\leqξ$（其中$ξ$为固定且不接近1的常数）。与这些结果根本不同的是，我们证明了当$ξ_k\leqξ$（常数$ξ<1$且不接近1）、$ξ_k=1-O(\|r_k\|)$和$ξ_k=1-O(\|r_k\|^2)$时，使用MINRES的非精确RQI通常分别三次、二次和线性收敛。因此，新的收敛条件比以往宽松得多。该理论可用于设计实用的停止准则，以更有效地实现该方法。数值实验证实了我们的结果。
英文摘要：
For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow inner tolerance $ξ_k\geq 1$ at outer iteration $k$ and can be considerably weaker than the condition $ξ_k\leqξ<1$ with $ξ$ a constant not near one commonly used in literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES method for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a more refined analysis and establish a number of new convergence results. Let $\|r_k\|$ be the residual norm of approximating eigenpair at outer iteration $k$. Then all the available cubic and quadratic convergence results require $ξ_k=O(\|r_k\|)$ and $ξ_k\leqξ$ with a fixed $ξ$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that $ξ_k\leqξ$ with a constant $ξ<1$ not near one, $ξ_k=1-O(\|r_k\|)$ and $ξ_k=1-O(\|r_k\|^2)$, respectively. Therefore, the new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.
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