大规模代数特征值高效迭代算法的构造及其应用

大规模代数特征值问题在科学与工程计算领域有着广泛应用，本项目拟研究大规模代数特征值迭代算法的构造及其应用。首先，提出求解内部特征值的增广Krylov子空间方法并给出收敛性分析；其次，将求解单个特征值的过滤的类Krylov序列方法推广至块形式以求解对称矩阵的多个特征值；再次，提出求解非对称矩阵特征值的Chebyshev-Davidson方法，根据特征值的分离情况讨论其收敛性质，并将其推广至求解多个特征值的块Chebyshev-Davidson方法；然后，改进预处理的Lanczos方法，并给出调谐预处理子下该方法的局部三次收敛性分析；进而，研究当前流行迭代方法的全局收敛性分析；最后，基于Google矩阵的特殊结构，我们将特征值求解算法应用到PageRank问题中。

Large algebraic eigenvalue problems arise widely in scientific computing and engineering applications. This research project aims to investigate the construction of efficient iteration algorithms with its applications for large eigenvalue problems. Firstly, we would like to construct the augmented Krylov subspace method for computing interior eigenvalues, and present the corresponding convergence analysis; Secondly, we generalize the filtered Krylov-like sequence method to block form to seek several eigenvalues of symmetric matrices; Thirdly, we present the Chebyshev-Davidson method for non-symmetric eigenvalue problems, investigate the convergence property in terms of the distribution of eigenvalues, and, in order to compute more eigenvalues, we give the corresponding block Chebyshev-Davidson method; Besides, we improve the preconditioned Lanczos method, and show its locally cubic convergence with tuned preconditioners; Moreover, we focus on the global convergence analyses of some state-of-the art iteration methods; Finally, based on the elegant structure of the Google matrices, we apply some iterative methods for eigenvalue problems to the computation of PageRank.