Lanczos algorithm for the quadratic eigenvalue problem in engineering applications

Abstract

A new approach of employing the Lanczos two-sided recursion to solve the quadratic eigenvalue problem is presented. The methodology employed retains the n order quadratic problems as posed, without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigenvalue problem of order 2 n. Pursuing the concept of finding a ‘characteristic matrix’ to yield an ‘eigenvalue square matrix’, the Lanczos recursion has been devised to solve the quadratic eigenvalue problem. Appropriate proofs showing the Lanczos biorthogonal transformation are included. The example problems presented validate the algorithm and show the effectiveness of the method for large quadratic eigenproblems.