It is well-known that the finite difference discretization of the Laplacian eigenvalue problem $-\Delta u = \lambda u$ leads to a matrix eigenvalue problem (EVP) $A x= \lambda x$ where the matrix $A$ is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \cite{strang2014functions}. We generalize the results and develop analytical solutions to the generalized matrix eigenvalue problems (GEVPs) $A x= \lambda Bx$ which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.
翻译:众所周知,Laplacian egenvaly问题($- delta u =\ lambda u 美元)的有限差异分解导致一个基质元素值问题(EVP) Ax= lambdax$A 美元x= lambdax$美元,其中矩阵值为Toeplitz+Hankel。 Strang 和 MacNamara +cite{strang2014函数}给出了具有不同边界条件的三对角矩阵的分析解决方案。我们推广了结果,并为通用基质值问题(GVPPPs)制定了分析解决方案。在本文中,我们侧重于为普通基值问题找到分析方法(GPEVS),而我们为普通基值问题找到分析方法。