Partial derivatives of the eigen-triplet of the quadratic eigenvalue problem depending on several parameters

摘要

This paper concerns the partial derivatives of the eigen-triplet of the quadratic matrix polynomial Q(p,λ)=λ2M(p)+λC(p)+K(p), where M(p),C(p),K(p)∈Cn×n are complex analytic matrix valued functions, p∈Cm is a complex parameter vector. First, the analyticity theorem for simple eigenvalues and the corresponding eigenvectors is given. Second, a new method is proposed to compute partial derivatives of the eigen-triplet. The derivatives of the eigen-triplet can be obtained by solving algebraic linear equations of order (n-1), where it only requires the information of the eigen-triplet whose partial derivatives are to be computed, and what is more important, the condition numbers of the coefficient matrices are “better” than those of the nonsingular coefficient matrices arisen in the bordered matrix method [1] and Nelson’s method [5]. Numerical tests show the feasibility and efficiency of the new method. The results are better or at least comparable with current methods.