An eigenspace method for computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems

Abstract

This paper concerns computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of the quadratic matrix polynomial Q(p,λ)=λ2M(p)+λC(p)+K(p) at p=p⁎. Computing derivatives of eigenvectors usually requires solving a certain singular linear system by transforming it into a nonsingular one. However, the coefficient matrix of the transformed linear system might be ill-conditioned. In this paper, we propose a new method for computing these derivatives, where the condition number of the coefficient matrix is the ratio of the maximum singular value to the minimum nonzero singular value of Q(p⁎,λ(p⁎)), which is generally smaller than those in current literature and hence leads to higher accuracy. Numerical examples show the feasibility and efficiency of our method.