Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues

Abstract

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2, …, λr of the matrix satisfy |λ1| ⩽ … ⩽ |λr| and |λs| < |λs+1| (s ⩽ r−1), then associated with any eigenvalue λi (i ⩽ s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi/λs+1|q+1, where the approximate method only uses the eigenpairs corresponding to λ1, λ2, …, λs. A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.