Eigensystem Computations

这个项目是关于计算一般矩阵的特征系统的算法的发展，通过将它们简化为比通常的海森伯格形式更紧凑的形式，并通过将LR算法应用于压缩形式。初步测试表明，该算法比标准QR算法在更短的时间内得到准确的结果。对于三对角线矩阵的最精简形式，将开发用于并行计算机的分治算法。对于需要求解代数Riccati方程的特化Hamilton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.