Nonlinear Rayleigh functionals

Rayleigh functionals are the generalization of Rayleigh quotients for matrices to nonlinear eigenvalue problems. While analyzing the state of the art we address some problems concerning the existing definitions and give different new ones. These new definitions are more general in that they allow complex eigenvalues and operators. However, in order to obtain local existence and uniqueness, we need to restrict to target eigentriplets (λ∗,x∗,y∗) with right and left normalized eigenvectors x∗,y∗ such that and to sufficiently good eigenvector approximations defining the Rayleigh functional. As there is a one-sided and a two-sided Rayleigh quotient, we introduce the one-sided and the two-sided Rayleigh functional. We estimate the quality of the Rayleigh functional in terms of the angles between target eigenvectors and their approximations and derive perturbation bounds and first order perturbation results of the same kind and order as known for linear problems. We show that stationarity holds in the same way. We conclude with an analysis of the so called generalized Rayleigh quotient proposed by Lancaster for matrix polynomials. We study the connection between this generalized Rayleigh quotient and the two-sided Rayleigh functional, and compare another variant without left eigenvector approximations with the one-sided Rayleigh functional.