Accelerated Iterative Procedure for Calculating Eigenvector Derivatives

Abstract

An iterative method for the computation of eigenvector derivatives of real-valued, symmetric systems has been proposed. The process is shown to converge to the exact solution with any initial values, and it is numerically stable. However, since the convergence rate is determined by the ratios of the eigenvalues, the convergence rate will be prohibitively slow when the ratio is close to 1. Two ef® cient accelerated algorithms are presented. Numerical examples show that when the eigenvalue ratio is greater than 0 .70, the computational effort in the iterative process is drastically reduced. The procedure incorporates the method of frequency shift to deal with a singular stiffness matrix that also provides an additional improvement for the convergence. When the eigenvalue ratios are extremely close to 1, the second generalized stiffness matrix inverse is suggested to reestablish a high convergence rate. The procedure can be used as an exact as well as an approximate method. It requires no more eigenvectors than those whose derivatives are to be calculated and can be applied to systems with repeated eigenvalues.