Journal Article10.1007/BF01390054
Homotopy algorithm for symmetric eigenvalue problems
Tien-Yien Li,Noah H. Rhee +1 more
51
TL;DR: The numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results shows that the homOTopy method can be very efficient especially for graded matrices.
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Abstract: The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.
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Citations
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Introduction to Numerical Continuation Methods
Eugene L. Allgower,Kurt Georg +1 more
- 01 Jan 1987
TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
A General Approach to Obtain Series Solutions of Nonlinear Differential Equations
Shijun Liao,Y Tan +1 more
TL;DR: Based on homotopy, which is a basic concept in topology, a general analytic method was proposed to obtain series solutions of nonlinear differential equations in this article, where the authors showed that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)thorder linear differential equations, where κ≥ 1 can be any a positive integer.
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Solving eigenvalue problems of real nonsymmetric matrices with real homotopies
TL;DR: In this article, a homotopy continuation algorithm for solving eigenvalue problems of real nonsymmetric matrices is developed based on divide and conquer strategy, which makes most of the eigen-paths almost straight lines and extremely easy to follow.
49
References
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Josef Stoer,Roland Bulirsch +1 more
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TL;DR: This well written book is enlarged by the following topics: B-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for theLR and QR algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations and preconditioning techniques.
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The Symmetric Eigenvalue Problem
Beresford N. Parlett
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TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
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The Symmetric Eigenvalue Problem.
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
3.4K
Studies of the Jahn-Teller effect .II. The dynamical problem
TL;DR: In this paper, the vibronic energy levels of a symmetrical nonlinear molecule in a spatially doubly degenerate electronic state which is split in first order by a doubly-degenerate vibrational mode are examined.
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