Research On Backward Error And Condition Number Analysis For Matrix Polynomials Solved By Fiedler Linearization

The polynomial eigenvalue problem is a typical nonlinear eigenvalue problem,which is widely used the control theory and structural engineering calculation.In this thesis,the backward error and condition number of polynomial eigenvalue problems are solved by Fiedler linearization and generalized Fiedler linearization,so as to measure the accuracy of solving polynomial eigenvalue problems by Fiedler linearization.In this thesis,we construct the upper bound of the ratio between the backward error of the eigenpair of the Matrix polynomial problem and the backward error of the Fiedler linearization,and establish the upper bound of the ratio between the Fiedler linearization eigenvalue condition number and Matrix polynomial eigenvalue condition number.By analyzing the upper bound,it is concluded that the norm size of the coefficient matrix is the main factor affecting the upper bound.Therefore,a kind of preprocessing technique is proposed to improve the backward error and condition number of Fiedler linearization method of solving polynomial eigenvalue problems.The preprocessing technology proposed in this thesis include the balance technology and the scaling technology.The backward error ratio bound and the condition number ratio bound of the Fiedler linearization method based on the preprocessing technique are constructed.The backward error ratio bound before and after the pretreatment technique is compared,and the condition number ratio bound before and after the pretreatment technique is compared.Theoretical analysis and numerical experiment verify that the preprocessing technology can improve the backward error and condition number of the Fiedler linearization method for solving Matrix polynomial eigenvalue problems.