Analytical solutions to some generalized and polynomial eigenvalue problems

Quanling Deng

doi:10.1515/spma-2020-0135

Abstract

AbstractIt is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu=λuleads to a matrix eigenvalue problem (EVP)Ax=λxwhere the matrixAis Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs)Ax=λBxwhich 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.

Highlights

It is well-known that the following tridiagonal Toeplitz matrix − A =n×n has analytical eigenpairs with xj =T where λj = − cos(jπh), xj,k = c sin(jπkh), h = n +, j, k =, · · ·, n with ≠ c ∈ R; see, for example, [20, p. 514] for a general case of tridiagonal Toeplitz matrix
We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the nite element method (FEM) and isogeometric analysis (IGA)
It is well-known that the following tridiagonal Toeplitz matrix n×n has analytical eigenpairs with xj =T where λj = − cos(jπh), xj,k = c sin(jπkh), h = n +, j, k =, · · ·, n with ≠ c ∈ R; see, for example, [20, p. 514] for a general case of tridiagonal Toeplitz matrix

Summary

Introduction

It is well-known that the following tridiagonal Toeplitz matrix n×n has analytical eigenpairs (λj , xj) with xj = (xj, , · · · , xj,n)T where λj = − cos(jπh), xj,k = c sin(jπkh), h = n + , j, k = , , · · · , n with ≠ c ∈ R (we assume that c is a nonzero constant throughout the paper); see, for example, [20, p. Let n×n the GEVP Ax = λBx has analytical eigenpairs (λj , xj) with xj = (xj, , · · · , xj,n)T where (see [16, Sec. 4] for a scaled case; A is scaled by /h while B is scaled by h) λj = − + + cos(jπh) , xj,k = c sin(jπkh), h = n + , j, k = , , · · · , n.

Results

Conclusion