An Algorithm for Generalized Matrix Eigenvalue Problems

1973 ◽  
Vol 10 (2) ◽  
pp. 241-256 ◽  
Author(s):  
C. B. Moler ◽  
G. W. Stewart
Keyword(s):  
Eigenvalue Problems ◽  
Matrix Eigenvalue ◽  
Generalized Matrix ◽  
Matrix Eigenvalue Problems

scholarly journals Analytical solutions to some generalized and polynomial eigenvalue problems

2021 ◽  
Vol 9 (1) ◽  
pp. 240-256
Author(s):  
Quanling Deng
Keyword(s):  
Eigenvalue Problem ◽  
Analytical Solutions ◽  
Eigenvalue Problems ◽  
Hankel Matrices ◽  
Various Boundary Conditions ◽  
Tridiagonal Matrices ◽  
Matrix Eigenvalue ◽  
The Matrix ◽  
Generalized Matrix ◽  
Matrix Eigenvalue Problems

Abstract It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is 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 = 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.

scholarly journals Some Modified Matrix Eigenvalue Problems

SIAM Review ◽  
1973 ◽  
Vol 15 (2) ◽  
pp. 318-334 ◽  
Author(s):  
Gene H. Golub
Keyword(s):  
Eigenvalue Problems ◽  
Matrix Eigenvalue ◽  
Matrix Eigenvalue Problems

Real Asymmetric Matrix Eigenvalue Analysis for Dynamic Instability Problems

2000 ◽  
Author(s):  
Heewook Lee ◽  
Noboru Kikuchi
Keyword(s):  
Control Systems ◽  
Eigenvalue Problems ◽  
Dynamic Instability ◽  
Eigenvalue Analysis ◽  
Complex Eigenvalue ◽  
Friction Forces ◽  
Matrix Eigenvalue ◽  
Complex Eigenvalues ◽  
Complex Eigenvalue Analysis ◽  
Matrix Eigenvalue Problems

Abstract Complex eigenvalue analysis is widely used when the dynamic instability of the structure is in doubt due to friction forces, aerodynamic forces, control systems, or other effects. MSC/NASTRAN upper Hessenberg method and MATLAB eigenvalue solver produce fictitious nonzero real parts for real asymmetric matrix eigenvalue problems. For dynamic instability problems, since nonzero real parts of complex eigenvalues determine the unstable eigenvalues, the accuracy of real parts becomes crucial. The appropriate double shift QR or the QZ algorithms are applied to eliminate fictitious nonzero real parts and produce only authentic complex eigenvalues for real asymmetric matrix eigenvalue problems. Numerical examples are solved using the double shift QR and the QZ algorithms, and the results are compared with the results of MSC/NASTRAN upper Hessenberg method and MATLAB solvers.

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