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

作者：Lu Xin^*; Xu Shu fang

来源：Applied Numerical Mathematics, 2015, 94: 201-208.

DOI：10.1016/j.apnum.2015.03.011

• Derivatives of eigenvalues and eigenvectors
• Quadratic eigenvalue problem
• Eigenspace
• Semi-simple eigenvalue

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

• 出版日期2015-8
• 单位北京大学; 北京应用物理与计算数学研究所