Nonlinear Eigenvalue Problems

Nonlinear eigenvalue problems arise in many fields of naturaland engineering sciences. Theoretical and practical results arescattered in the literature and in most cases they have beendeveloped for a certain type of problem. In this book we considerthe most general nonlinear eigenvalue problem without assumptionson the structure or spectrum and provide basic facts. NonlinearRayleigh functionals are analyzed in detail and in different forms.New methods for the computation of eigenvalues and -vectors aredesigned based on Rayleigh functionals, and well-knownJacobi--Davidson methods are discussed. Asymptotically cubicconvergence of the two-sided nonlinear Jacobi--Davidson method isshown. The special case of nonlinear complex symmetric eigenvalueproblems is examined. We show the appropriate definition of acomplex symmetric Rayleigh functional, which is used to derive acomplex symmetric Rayleigh functional iteration which convergeslocally cubically, and the complex symmetric residual inverseiteration method. All methods are also illustrated numerically. Thebook is addressed to mathematicians as well as engineers interestedin nonlinear eigenvalue problems. Nonlinear eigenvalue problems arise in many fields of natural and engineering sciences. Theoretical and practical results are scattered in the literature and in most cases they have been developed for a certain type of problem. In this book we consider the most general nonlinear eigenvalue problem without assumptions on the structure or spectrum and provide basic facts.§Nonlinear Rayleigh functionals are analyzed in detail and in different forms. New methods for the computation of eigenvalues and -vectors are designed based on Rayleigh functionals, and well-known Jacobi--Davidson methods are discussed. Asymptotically cubic convergence of the two-sided nonlinear Jacobi--Davidson method is shown.§The special case of nonlinear complex symmetric eigenvalue problems is examined. We show the appropriate definition of a complex symmetric§Rayleigh functional, which is used to derive a complex symmetric Rayleigh functional§iteration which converges locally cubically, and the complex symmetric residual inverse iteration method. All methods are also illustrated numerically.§The book is addressed to mathematicians as well as engineers interested in nonlinear eigenvalue problems.