A Continuation Approach to Eigenvalue Problems

本课题将研究基于同伦延拓思想的新特征值算法。这包括将一个简单矩阵变形为所要讨论的矩阵，同时将简单矩阵的容易找到的特征对跟踪到所要寻找的特征对。要研究的问题包括：-进一步探索最近开发的对称三对角矩阵特征值问题的延拓算法。-大型稀疏矩阵特征值问题的延拓算法。-并行化和矢量化这些算法为各种先进的架构。-非对称矩阵特征值问题的延拓算法。重点将放在高效计算算法的发展及其数值实现上。有希望的是，这项工作将导致软件包，可以应用于实际的特征值问题，特别是那些大的，稀疏矩阵。

This project will investigate new eigenvalue algorithms based upon the idea of homotopy continuation. These consist of deforming a simple matrix into the matrix in question, while following the easily found eigenpairs of the simple matrix to the eigenpairs being sought. The issues to be studied include: - Further exploration of a recently developed continuation algorithm for eigenvalue problems of symmetric triadiagonal matrices. - Continuation algorithms for eigenvalue problems of large sparse matrices. - Parallelization and vectorization of such algorithms for a variety of advanced architectures. - Continuation algorithms for eigenvalue problems of nonsymmetric matrices. The emphasis will be on the development of efficient computational algorithms and their numerical implementation. Hopefully, the work will lead to software packages that can be applied to practical eigenvalue problems, especially those of large, sparse matrices.