Condition Numbers Of A Semi-simple Eigenvalue Of A Quadratic Eigenvalue Problem Based On Sensitivity Analysis Of Eigenpairs

The condition numbers of eigenvalues of matrices measure the sensitivities of eigenvalues to small perturbation of matrix.They're important on assessing the quality of numerical algorithms for eigenvalue problems.This paper considers the condition number of multiple eigenvalue of regular quadratic eigenvalue problem.Based on the properties of multiple eigenvalue of quadratic eigenvalue problem analytically dependent on several parameters,we give various definitions for condition numbers of semi-simple eigenvalue of regular quadratic eigenvalue problem.Utilizing SVD and the properties of unitarily invariant norm,we derive the computational expressions for the introduced condition numbers.Compared with the existing condition numbers of multiple eigenvalues of quadratic eigenvalue problem,the condition numbers defined in this paper can measure not only the worst case sensitivity of semi-simple eigenvalue,but also the different sensitivities of the eigenvalues spawned from the semi-simple eigenvalue.On the other hand,we consider the ill-posed perturbation problem of semi-simple eigenvalue ?1,of quadratic eigenvalue problem,and an upper bound on the perturbation of quadratic eigenvalue problem such that the perturbed quadratic eigenvalue problem having ?1 with larger multiplicity is given.