Eigensystem Computations

这个项目致力于通过将一般矩阵的特征系统简化为比通常的Hessenberg形式更紧凑的形式并应用凝聚形式的LR算法来发展计算一般矩阵的特征系统的算法。初步测试表明，对于稠密特征系统，该算法将在比标准QR算法更短的时间内得到准确的结果。对于最浓缩的三对角矩阵形式，将开发一种用于并行计算机的分而治之算法。对于需要解代数Riccati方程的特殊的哈密顿特征值问题，我们的算法将基于保持矩阵哈密顿结构的辛相似变换。

This project is concerned with the development of algorithms for calculating the eigensystems of general matrices by reducing them to a form which is more compact than the usual Hessenberg form and by applying the LR algorithm ot the condensed form. Preliminary tests suggest that the proposed algorithm will produce accurate results in a much shorter time than the standard QR algorithm for a dense eigensystem. For the most condensed form of a tridiagonal matrix, a divide-and-conquer algorithm will be developed for use in parallel computers. For a specialized Hamilton eigenvalue problem which requires the solution of an algebraic Riccati equation, our algorithm will be based on symplectic similarity transformations which preserve the Hamiltonian structure of the matrix.