Numerical Methods for the Unsymmetric Tridiagonal EigenvalueProblem

高阶本征值问题在许多应用中出现。 的 长的计算时间和大的存储需求，这些问题 推动快速串行和高效并行的发展 特征解算器 将研究新的数值方法， 计算所有真实的非对称的特征值和特征向量 三对角矩阵，出现在应用和减少 一般真实的矩阵 近期目标是提供新的方法 对于三对角的情况。 因为目前通用的方法 特征值问题可能会受到并行效率低下或数值 不稳定性，更大的目标是开发精确的并行方法 最终可能会扩展到一般的特征值问题， 初始化为三对角形式。 这项研究源于已经成功的方法， 对称三对角特征值问题 这项工作将从一项研究开始开始 的并行方法计算特征向量的非对称 给定其计算特征值的三对角矩阵。 它将继续 通过开发分治方法来计算两个特征值， 非对称三对角矩阵的特征向量 它将 最后得出一个机制，粗略定位任何特征值， 三对角矩阵 后一种技术适用于 用于计算特征值的加速求根方法。 串行 将设计、实现和测试并行算法 所有方法。

Large order eigenproblems arise in a variety of applications. The long computing times and large storage requirements of these problems motivate development of fast serial and efficient parallel eigensolvers. New numerical methods will be investigated for computing all eigenvalues and eigenvectors of real unsymmetric tridiagonal matrices that arise in applications and from reduction of general real matrices. The immediate goal is to provide new methods for the tridiagonal case. Because current methods for the general eigenproblem can suffer from parallel inefficiency or numerical instability, the larger goal is to develop accurate parallel methods that might ultimately be extended to the general eigenproblem without initial reduction to tridiagonal form. The research derives from methods that have been successful for the symmetric tridiagonal eigenproblem. The work will begin with a study of parallel methods for computing eigenvectors of an unsymmetric tridiagonal matrix given its computed eigenvalues. It will continue by developing divide and conquer methods to compute both eigenvalues and eigenvectors of an unsymmetric tridiagonal matrix. It will conclude with a mechanism for roughly locating the eigenvalues of any tridiagonal matrix. The latter technique is geared toward accelerating rootfinding methods for computing eigenvalues. Serial and parallel algorithms will be designed, implemented, and tested for all approaches.