Analytical solutions to some generalized and polynomial eigenvalue problems

Abstract It is well-known that the finite difference discretization of Laplacian eigenvalue problem ? ?u = ?u leads to a matrix (EVP) Ax ?x where A Toeplitz-plus-Hankel. Analytical solutions tridiagonal matrices with various boundary conditions are given in recent work Strang and MacNamara. We generalize results develop analytical certain generalized problems (GEVPs) ?Bx which arise from element method (FEM) isogeometric analysis (IGA). The FEM corner-overlapped block-diagonal while IGA almost In fact, correction Toeplitz-plus-Hankel gives better numerical method. this paper, we focus on finding eigenpairs GEVPs developing methods our motivation. also obtained for some polynomial (PEVPs). Lastly, eigenvector-eigenvalue identity (rediscovered coined recently EVPs) derive trigonometric identities.