Numerical deflation of the transcendental eigenvalue problem

Abstract

Structural dynamic analysis of continuous systems such as rods, beams or plates involves determination of eigenvalues of matrices with transcendental elements. Conventionally, the eigenvalues are found one-by-one with an iterative process where the initial guess governs the convergence to the individual solution. To find a range of eigenvalues it is necessary to repeat the process starting from different initial points in anticipation that new eigenvalues, which were not extracted previously, be found. However, convergence to such new eigenvalues may not be assured.It is shown that numerical deflation, in the form of adding singularities near the extracted eigenvalues, allows determination of a prescribed number of different eigenvalues from a fixed-point initial guess. The case of repeated eigenvalues where the process is ill conditioned is also analyzed. Examples demonstrate the various results.