A Continuation Approach to Eigenvalue Problems

本计画将探讨新的特徴值演算法 同伦延拓的概念 这些包括变形a 简单的矩阵到矩阵的问题，而下面的容易 找到了简单矩阵的特征对。 拟研究的问题包括： - 进一步探索最近开发的 特征值问题的延拓算法 对称三对角矩阵 - 大系统特征值问题的连续化算法 稀疏矩阵 - 这类算法的离散化和向量化 适用于各种高级架构。 - 特征值问题的延拓算法 非对称矩阵 重点将放在发展高效的 计算算法及其数值实现。 希望这项工作将导致可以应用的软件包 实际的特征值问题，特别是那些大，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.