Hybrid Algorithms for Solving the Algebraic Eigenvalue Problem with Sparse Matrices

2017 ◽  
Vol 53 (6) ◽  
pp. 937-949 ◽  
Author(s):  
A. N. Khimich ◽  
A. V. Popov ◽  
O. V. Chistyakov
Keyword(s):  
Eigenvalue Problem ◽  
Sparse Matrices ◽  
Hybrid Algorithms ◽  
Algebraic Eigenvalue Problem

The variation method and the algebraic eigenvalue problem

1969 ◽  
Vol 3 (5) ◽  
pp. 711-722 ◽  
Author(s):  
A. Wallis ◽  
D. L. S. McElwain ◽  
H. O. Pritchard
Keyword(s):  
Eigenvalue Problem ◽  
Variation Method ◽  
Algebraic Eigenvalue Problem

scholarly journals A Convenient Method to Obtain Stellar Eigenfrequencies

1983 ◽  
Vol 66 ◽  
pp. 331-341
Author(s):  
M. Knölker ◽  
M. Stix
Keyword(s):  
Eigenvalue Problem ◽  
Model Comparison ◽  
Symmetric Matrix ◽  
Outer Boundary ◽  
Solar Model ◽  
Pressure Perturbation ◽  
Stellar Oscillations ◽  
The Matrix ◽  
First Results ◽  
Algebraic Eigenvalue Problem

AbstractThe differential equations describing stellar oscillations are transformed into an algebraic eigenvalue problem. Frequencies of adiabatic oscillations are obtained as the eigenvalues of a banded real symmetric matrix. We employ the Cowling-approximation, i.e. neglect the Eulerian perturbation of the gravitational potential, and, in order to preserve selfadjointness, require that the Eulerian pressure perturbation vanishes at the outer boundary. For a solar model, comparison of first results with results obtained from a Henyey method shows that the matrix method is convenient, accurate, and fast.

A Separation Principle for Gyroscopic Conservative Systems

1997 ◽  
Vol 119 (1) ◽  
pp. 110-119 ◽  
Author(s):  
L. Meirovitch
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Separation Theorem ◽  
Separation Principle ◽  
Numerical Illustration ◽  
Closed Form Solutions ◽  
Conservative Systems ◽  
Algebraic Eigenvalue Problem

Closed-form solutions to differential eigenvalue problems associated with natural conservative systems, albeit self-adjoint, can be obtained in only a limited number of cases. Approximate solutions generally require spatial discretization, which amounts to approximating the differential eigenvalue problem by an algebraic eigenvalue problem. If the discretization process is carried out by the Rayleigh-Ritz method in conjunction with the variational approach, then the approximate eigenvalues can be characterized by means of the Courant and Fischer maximin theorem and the separation theorem. The latter theorem can be used to demonstrate the convergence of the approximate eigenvalues thus derived to the actual eigenvalues. This paper develops a maximin theorem and a separation theorem for discretized gyroscopic conservative systems, and provides a numerical illustration.

The Algebraic Eigenvalue Problem

1966 ◽  
Vol 20 (96) ◽  
pp. 621
Author(s):  
E. I. ◽  
J. H. Wilkinson
Keyword(s):  
Eigenvalue Problem ◽  
Algebraic Eigenvalue Problem
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