关于使用Lanczos方法求解线性系统的不精确Rayleigh商迭代的收敛性
On Convergence of the Inexact Rayleigh Quotient Iteration with the Lanczos Method Used for Solving Linear Systems
AI总结：
针对使用Lanczos方法求解内线性系统的不精确Rayleigh商迭代，建立了新的局部二次收敛理论，放宽了传统收敛条件，并推广到调谐预条件Lanczos方法。
摘要：
对于Hermitian不精确Rayleigh商迭代（RQI），作者建立了新的局部一般收敛结果，独立于内线性系统的迭代求解器。理论表明，在新条件（称为一致正定性条件）下，该方法局部二次收敛。本文首先考虑使用无预条件Lanczos方法求解线性系统的不精确RQI的局部收敛性。推导了Lanczos方法得到的线性系统残差（其范数为$\xi_{k+1}$）的一些吸引性质。基于这些性质和新的一般收敛结果，我们进行了精细分析并建立了新的局部收敛结果。证明当$\xi_{k+1}\leq\xi$且常数$\xi\geq 1$时，使用Lanczos的不精确RQI二次收敛。当$\xi_{k+1}$被当前近似特征对残差范数$\|r_k\|$的倒数的小倍数有界时，该方法保证线性收敛。这些结果与始终要求$\xi_{k+1}<1$的现有收敛结果根本不同，并对方法的有效实现有重要影响。我们将新理论推广到使用调谐预条件Lanczos求解线性系统的不精确RQI。基于新理论，我们可以设计实用准则来控制$\xi_{k+1}$以实现二次收敛，并比以前更有效地实现该方法。数值实验证实了我们的理论。
英文摘要：
For the Hermitian inexact Rayleigh quotient iteration (RQI), the author has established new local general convergence results, independent of iterative solvers for inner linear systems. The theory shows that the method locally converges quadratically under a new condition, called the uniform positiveness condition. In this paper we first consider the local convergence of the inexact RQI with the unpreconditioned Lanczos method for the linear systems. Some attractive properties are derived for the residuals, whose norms are $ξ_{k+1}$'s, of the linear systems obtained by the Lanczos method. Based on them and the new general convergence results, we make a refined analysis and establish new local convergence results. It is proved that the inexact RQI with Lanczos converges quadratically provided that $ξ_{k+1}\leqξ$ with a constant $ξ\geq 1$. The method is guaranteed to converge linearly provided that $ξ_{k+1}$ is bounded by a small multiple of the reciprocal of the residual norm $\|r_k\|$ of the current approximate eigenpair. The results are fundamentally different from the existing convergence results that always require $ξ_{k+1}<1$, and they have a strong impact on effective implementations of the method. We extend the new theory to the inexact RQI with a tuned preconditioned Lanczos for the linear systems. Based on the new theory, we can design practical criteria to control $ξ_{k+1}$ to achieve quadratic convergence and implement the method more effectively than ever before. Numerical experiments confirm our theory.
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