On Interval Estimates Of Perturbations Of Generalized Eigenvalues For Diagonalizable Pairs

The generalized eigenvalues of diagonalizable matrix pairs have a great effect on large-scale processing data such as cybernetics,theoretical physics,statistics,operations research,and image processing.For a matrix that can be diagonalized,when calculating its generalized eigenvalues,due to the existence of errors,there is a certain deviation between the numerical solution of the generalized eigenvalues and the exact value,so it is necessary to estimate the correctness of the calculation results..This requires studying the perturbation of the generalized eigenvalues of the diagonalizable matrix pairs,which leads to the interval estimation problem of the perturbations of the generalized eigenvalues of the diagonalizable matrices.Many researchers have also been studying the subject of perturbation of generalized eigenvalues by diagonalization matrices.In this article,we study this topic from a new perspective.Based on the distance and properties of two points in the Grassmann manifold,we can obtain more accurate upper and lower bounds that can perturb the generalized eigenvalues of the diagonalized matrix.The main results improve the existing results of Chen(2007)[1] and numerical examples illustrate the efficiency of the theorectical results.The specific work done in this article is as follows:The first chapter mainly introduces the research background and current status of the thesis and what needs to be studied in this article.The second chapter mainly gives some necessary symbols and definitions,as well as some existing conclusions,and some useful lemmas that prove the main theorems in the third part.Chapter 3 gives more accurate upper bounds and interval estimates for generalized eigenvalue perturbations of diagonalizable matrix pairs.Chapter 4 gives numerical examples to prove the theoretical results of Chapter 3.Chapter 5 is a summary of the whole paper.