Computations of Nonsymmetric Eigenvalue Problems and the Generalized Singular Value Decomposition

随着先进计算设备的可用性和浮点运算环境的改善，对于许多数值线性代数问题，有进一步开发和分析精确、稳定和高效的数值解的需要和机会。本文研究了大型非对称矩阵的特征值计算、具有特定谱的不变子空间的计算、非对称特征值问题的条件数估计、广义QR分解和广义奇异值分解的计算等基本矩阵计算问题。研究的目标是为高性能计算机开发高效的算法和便携式软件，这些问题在科学计算和其他数值分析领域的许多应用激发了研究的动机。

With the availability of advanced computing facilities and the improvement of the environment of floating point arithmetic, there is a need and an opportunity for further development and analysis of accurate, stable and efficient numerical solutions for many numerical linear algebra problems. This proposal is concerned with several fundamental matrix computation problems, such as computing eigenvalues of large nonsymmetric matrices, computing invariant subspaces with specific spectra, estimating condition numbers of the nonsymmetric eigenvalue problem, and computing the generalized QR factorization and the generalized singular value decomposition. The goal of the research is to develop efficient algorithms and portable software for high- performance computers, motivated by many applications of these problems in scientific computing and other areas of numerical analysis.