AI 中文总结
本文针对无再正交化的对称Lanczos算法开展有限精度稳定性分析，推导相关恒等式与定位关系，证明Greenbaum型向后稳定性结果，简化了Paige与Greenbaum的证明，仅在迭代次数中隐藏多项式因子。
AI 中文摘要
我们给出了对称Lanczos算法在无再正交化下的自包含有限精度分析，具体推导了扰动三项递推、Paige的正交性损失恒等式、所有计算Ritz值的包含关系及稳定Ritz值的定位，随后证明了Greenbaum型向后稳定性结果，展示了存在邻近问题使得精确Lanczos生成计算出的三对角矩阵。我们的证明简化了Paige和Greenbaum的证明，代价是在迭代次数中隐藏了多项式因子。
英文摘要
We give a self-contained finite-precision analysis of the symmetric Lanczos algorithm without reorthogonalization. In particular, we derive the perturbed three-term recurrence, Paige's loss-of-orthogonality identity, containment of all computed Ritz values, and localization of stabilized Ritz values. We then prove a Greenbaum-type backward stability result, exhibiting a nearby problem on which exact Lanczos produces the computed tridiagonal matrix. Our proofs simplify those of Paige and Greenbaum, at the cost of hiding polynomial factors in the iteration count.
