Second Order Numerical Differentiation Of The Quadratic Eigenvalue Problem

Quadratic eigenvalue problems are widely used in structural mechanics and other research fields.Numerical differentiation of quadratic eigenvalue problems has important applications in damage detection and model modification.This paper mainly studies the second order numerical differentiation of the quadratic eigenvalue problem,including the numerical calculation of the second order partial derivatives of the semi-single eigenvalue and its corresponding right and left eigenvectors,and analyzes the rounding error of the second order partial derivatives of the semi-single eigenvalue in this problem.In general,there are specific formulas to calculate the derivatives of eigenvalues,so it is easy to calculate,while it is more difficult to calculate the derivatives of eigenvectors.Among many methods,the eigenspace method is a better algorithm to calculate the semisimple eigenvalue and its corresponding eigenvector derivatives.Its main idea is to represent the columns of the eigenvector corresponding to the semisimple eigenvalue as a linear combination of basis vectors.And the basis vectors are derived from the basis vectors of the eigenspace corresponding to the semisimple eigenvalues whose derivatives are to be computed using some simple transformations.This method only needs to collect few information to achieve stable results.In consideration of the superiority of the eigenspace method,this paper extends its idea to the calculation of the second order partial derivatives of semisimple eigenvalues and their corresponding right-left eigenvectors in the quadratic eigenvalue problem,and gives the derivation process and the corresponding algorithm.Numerical test results show that the new algorithm has low error,high accuracy and stability.This article also analyzes rounding errors of second order partial derivatives of semisimple eigenvalues of the quadratic eigenvalue problems,and results in that the upper bound only depends on the coefficient matrix,eigenvalue and eigenvector,and the size of the problem.