Backward Errors And Condition Numbers For A Class Of Skew-Symmetric Eigenvalue Problems

Backward errors and condition numbers play important roles in modern numerical linear algebra: backward errors reveal the stability of a numerical method, condition numbers characterize the sensitivity of the solution of a problem to perturbations in data. They are both important tools to judge the quality of a computed solution. The error in a computed solution, backward error, and condition number are related by the inequality (correct to first order in the backward error): error <_~ condition number×backward error.For the actual background, given matrices usually have some special structure, when analyzing the solutions of structured problems, it is more suitable to use the following inequality: structured error <_~ structured condition number×structured backward error. This thesis considers the following eigenvalue problem:The problem is derived from geodesic computation on the Stiefel manifold. This thesis deals with structured backward errors and condition numbers for such skew-symmetric eigenvalue problems. The doubly structures (skew-symmetric structure and zero-block structure) are taken into consideration.This thesis consists of three sections:In Section 1, we summarize historical developments of condition number and backward error analysis, the present progress of the research on structured backward errors, structured condition numbers on the structured eigenvalue problems. Furthermore, we outline the content of the thesis.In Section 2, we derive computable formulas for the doubly structured normwise backward error in real case, and compare the results with the single structured case (skew-symmetric structure). Numerical results demonstrate that the ratio of doubly structured backward error and the single structured one can be large.In section 3, we derive computable formulas for the doubly structured condition number in complex case, and bounds for the doubly structured condition number in real case.