The Optimal Backward Perturbation Analysis For Orthogonal Matrix Eigenvalue Problem And Rice Condition Numbers Of Certain Characteristic Subspaces

The optimal backward perturbation analysis is a fruitful method developed in the last ten years. In general, a computed solution obtained by numerical methods is a approximate solution of the original problem. The backward error and the structured backward error of the approximate solution are the criteria to judge the stability and the strong stability of the numerical algorithm. Condition number is a measure of the sensitivity to the approximate solution for the perturbation of original date. The optimal backward error and the condition number are important measures to evaluate the quality of computed solution.This paper is composed of three chapters.In the first chapter, the background of the optimal backward perturbation theories are introduced, and the historical development of the research on the eigenvalue problems are summarized. The main results of the optimal backward perturbation theories on eigenvalue problems are outlined.In the second chapter, we deal with two types of the problem: real and complex, and analyzes the backward perturbation of eigenpair of orthogonal matrix. The expression or evaluation to the optimal structure backward error are given, compared with the result of optimal backward error. In addition, the backward error for eigenvalue and eigenvector are analyzed respectively.In the third chapter, the perturbation of invariant subspace, singular subspaces and deflating subspaces are discussed. According to Rice's idea, Rice condition numbers for these subspaces are defined by applying perturbation expansions of orthogonal projection operators. The expressions of the Rice condition numbers and the first-order perturbation estimations for these subspaces are derived. Compared with Sun's approach, the Rice condition numbers are independent of the choices of basis.