Some Doubly Strucured Matrix Eigenvalue Problems

Structured matrix eigenvalue problems arise in a number of practical applications, such as Hamiltonian matrix eigenvalue problem which still an open question so far. It is important that how to develop structure- preserving algorithms for eigenvalue problems using these special structures which usually reflect the physical background. Meanwhile, another significant problem is that how to measure the stability of these structure-preserving algorithms. There is much appraisal for these structure-preserving algorithms, in which backward error is very important. As structure-preserving algorithms to be concerned, the problems turn to be more complicated. Disturbed structured matrix is still restricted to the subclass that it originally belongs to, therefore, more exactly speaking, we should study structured backward error while studying structure-preserving algorithm.In this paper, we study structure-preserving algorithms for some doubly structured matrices and structured backward errors by Van Loan's squared reduced method, quasi-QR factorization and Wielandt-Hoffman theorem. The main results state as follows.1. We develop SR method further to structured Hamiltonian matrices of complex case and real structured symplectic matrix eigenvalue problems.2. Using quasi-QR factorization and Wielandt-Hoffman theorem, we study backward errors for five classes of doubly structured matrix problems raised by Tisseur[23] and upper and lower bounds easily to be computed for these structured backward errors.Finally, we point out some problems to be further discussed after summarizing the works done in this thesis.