论文标题
某些广义和多项式特征值问题的分析解决方案
Analytical solutions to some generalized and polynomial eigenvalue problems
论文作者
论文摘要
众所周知，拉普拉斯特征值问题$-ΔU=λu$的有限差异化导致矩阵特征值问题（EVP）$ a x =λx$，其中矩阵$ a $ a $ is toeplitz-plus-plus-hankel。在strang和macnamara \ cite {strang2014functions}中给出了具有各种边界条件的三角形矩阵的分析解决方案。我们将结果推广并为广义矩阵特征值问题（GEVPS）$ a x =λbx$开发分析解决方案，该解决方案是由有限元方法（FEM）和等化分析（IGA）产生的。 FEM矩阵是拐角处的块状型基因，而IGA矩阵几乎是toeplitz-plus-hankel。实际上，具有校正的IgA导致toeplitz-plus-hankel矩阵提供了更好的数值方法。在本文中，我们专注于找到GEVP的分析特征，同时开发更好的数值方法是我们的动机。还获得了某些多项式特征值问题（PEVP）的分析溶液。最后，我们概括了GEVP的特征向量 - 元素值身份（最近为EVP重新发现和创造），并得出一些三角体性。
It is well-known that the finite difference discretization of the Laplacian eigenvalue problem $-Δu = λu$ leads to a matrix eigenvalue problem (EVP) $A x= λx$ 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) $A x= λBx$ 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.
