The Study Of Numerical Methods And Backward Error Analysis Of Quadratic Eigenvalue Problem

The quadratic eigenvalue problem(QEP)is worth discussing in numerical algebra,and the application background of QEP is very extensive.So far,many scholars have conducted a lot of research on this problem and proposed various solutions.In practical applications,only a single or partial eigenvalue of QEP is required.The Sakurai-Sugiura with Rayleigh-Ritz procedure(SS-RR)method is a very effective method for solving partial eigenvalues.It projects the original QEP into a small to medium-sized eigenvalue problem.As for the solution of the small to medium-size QEP,a common method is to convert it into a generalized eigenvalue problem via linearization.We will introduce the SS-RR method with linearization to solve QEP in this paper.At the same time,considering that the backward stable algorithm can give an approximate accurate solution,the backward error is an important indicator for evaluating the stability of the numerical method.Therefore,we mainly study the following two contents.First,we analyze the backward error of SS-RR method with linearization when solving QEP.By combining the distribution of eigenvalues with the coefficient matrix norm of the projected QEP,we establish a tight upper bound on the backward error of the eigenvalue pair of original QEP.This upper bound provides effective information for analyzing the numerical stability of SS-RR method.Second,considering that SS-RR method may have numerical instability in solving some problems.Balancing is a commonly used and mature technique to improve the sensitivity of eigenvalues.This method improves the accuracy of backward error of required eigenvalues by reducing the norm of the coefficient matrix.This paper introduces the diagonal scaling technique proposed by Betcke,and integrates the diagonal scaling technique into the SS-RR method with linearization.Using this method to solve QEP,we have verified theoretically and experimentally that balancing processing can improve the backward error of eigenvalue pairs.