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

摘要

This paper concerns computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of the quadratic matrix polynomial Q(p,λ) = λ~2M(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.