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.
