从优化视角看特征值计算：论Jacobi-Davidson、IIGD、RQI与Newton更新
Eigenvalue Computation from the Optimization Perspective: On Jacobi-Davidson, IIGD, RQI, and Newton Updates
AI总结：
本文从优化视角揭示特征值计算与Newton法及子空间法的联系，提出新Newton更新，简化Jacobi-Davidson和IIGD方法，证明其子空间包含RQI方向，保留并增强局部二次收敛性，数值实验支持理论。
摘要：
我们讨论了特征值计算与使用Newton方法和子空间方法的优化之间的紧密联系。从这一联系中，我们推导出一类新的Newton更新。新的更新公式与著名的Jacobi-Davidson方法相似。这种相似性导致了Jacobi-Davidson方法和逆迭代广义Davidson（IIGD）方法的简化版本。我们证明了由这些方法中的每一种的更新方向增强的投影子空间能够包含Rayleigh商迭代（RQI）方向。因此，RQI方法的局部二次（对于正规矩阵为三次）收敛速度得以保留，并通过子空间方法得到加强。该理论得到了大量数值结果的支持。对于大规模特征值问题，还简要讨论了预条件公式。
英文摘要：
We discuss the close connection between eigenvalue computation and optimization using the Newton method and subspace methods. From the connection we derive a new class of Newton updates. The new update formulation is similar to the well-known Jacobi-Davidson method. This similarity leads to simplified versions of the Jacobi-Davidson method and the inverse iteration generalized Davidson (IIGD) method. We prove that the projection subspace augmented by the updating direction from each of these methods is able to include the Rayleigh quotient iteration (RQI) direction. Hence, the locally quadratic (cubic for normal matrices) convergence rate of the RQI method is retained and strengthened by the subspace methods. The theory is supported by extensive numerical results. Preconditioned formulations are also briefly discussed for large scale eigenvalue problems.
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