Eigensystem Computations

该项目致力于通过将一般矩阵的特征系简化为比通常的Hessenberg形式更紧凑的形式并应用LR算法来计算一般矩阵的特征系。初步测试表明，该算法将产生准确的结果，在一个更短的时间比标准的QR算法的密集特征系统。对于三对角矩阵的最压缩形式，将开发用于并行计算机的分治算法。对于一个专门的汉密尔顿特征值问题，需要解决的代数Riccati方程，我们的算法将基于辛相似变换保持矩阵的哈密顿结构。

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.