On the solution of generalized non-linear complex-symmetric eigenvalue problems

This paper brings an attempt toward the systematic solution of the generalized non-linear, complex-symmetric eigenproblem ( K ₀−iω C ₁−ω² M ₁−iω³ C ₂−ω⁴ M ₂−···) ϕ = 0 , with real, symmetric matrices K ₀, C _j, M _j ε R^n×n, which are associated with the dynamic governing equations of a structure submitted to viscous damping, as laid out in the frame of an advanced mode superposition technique. The problem can be restated as ( K _(ω)−ω M _(ω)) ϕ = 0 , where K _(ω)= K and M _(ω)= M are complex-symmetric matrices given as power series of the complex eigenfrequencies ω, such that, if (ω, ϕ ) is a solution eigenpair, ϕ ^T M _(ω) ϕ =1 and ϕ ^T K _(ω) ϕ =ω. The traditional Rayleigh quotient iteration and the more recent Jacobi–Davidson method are outlined for complex-symmetric linear problems and shown to be mathematically equivalent, both with asymptotically cubic convergence. The Jacobi–Davidson method is more robust and adequate for the solution of a set of eigenpairs. The non-linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis, thus also presenting cubic convergence. Two examples help us to visualize some of the basic concepts developed. Three more examples illustrate the applicability of the proposed algorithm to solve non-linear problems, in the general case of underdamping, but also for overdamping combined with multiple and close eigenvalues. Copyright © 2007 John Wiley & Sons, Ltd.