The Partial Derivatives And Error Analysis Of The Eigen-pair Of The Quadratic Eigenvalue Problem

The vibration equations of many buildings,such as bridges and roads,are partial differential equations.These equations are discretized into second order differential equations by the finite element method,and then by separating the variables,we can get the quadratic eigenvalue problem(QEP).This paper mainly studies the error analysis of the first order derivatives of both simple and semi-simple eigenvalues and the numerical calculation of the higher oeder derivative in the quadratic eigenvalue problem.The calculation of the derivative of the eigenvalue is generally relatively simple,and the formula is definite.However,due to the non-uniqueness of eigenvectors,it is relatively complicated to calculate the derivatives of eigenvectors.The single mode method is a good algorithm to calculate the derivative of the eigenvector corresponding to a simple eigenvalue.The basic idea of the single mode method is to represent the partial derivatives of the desired eigenvectors by a linear combination of a set of base vectors,where this set of base vectors is obtained by Householder transformation of the eigenvectors.The advantage of the algorithm is that it requires less information and is relatively stable.In this paper,we first analyze the forward round-off error of the partial derivatives of semi-simple eigenvalue,and obtain the upper bound of the round-off error of the partial derivatives of semi-simple eigenvalue of the nonsymmetric quadratic eigenvalue problem.This upper bound is determined only by the scale,coefficient matrices,eigenvalues and their corresponding eigenvectors of the quadratic eigenvalue problem.Numerical experiments show that this result is a good estimation.After that,we discuss perturbation analysis of the partial derivatives of the simple eigenvalue of the quadratic eigenvalue problem,and it is proved by the numerical tests that the calculation result is stable.Finally,based on the superiority of the single mode method,we extend the single mode method to the calculation of the third partial derivative of the simple eigen-triplet of the quadratic eigenvalue problem,and deduce the calculation formula and give the corresponding algorithm.Numerical experiments show that the single mode method is also applicable at this time,with high precision,small error and stability.