Journal Article10.1137/0914046
An algorithm for symmetric tridiagonal eigenproblems: divide and conquer with homotopy continuation
Kuiyuan Li,Tien-Yien Li +1 more
31
TL;DR: A new algorithm for finding all the eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix based on the homotopy continuation approach coupled with the strategy of “divide and conquer” is presented.
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Abstract: This paper presents a new algorithm for finding all the eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix. The algorithm is based on the homotopy continuation approach coupled with the strategy of “divide and conquer.” Evidenced by the numerical results, the algorithm given here provides a considerable advance over previous attempts to use the homotopy method for eigenvalue problems. Numerical comparisons of this algorithm with the methods in the widely used EISPACK library, as well as Cuppen’s divide and conquer method, are presented. It appears that the algorithm is strongly competitive in terms of speed, accuracy, and orthogonality. The performance of the parallel version of this algorithm is also presented. The natural parallelism of the algorithm makes it an excellent candidate for a variety of advanced architectures.
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Citations
Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem
TL;DR: The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices and may be a useful tool for determining the stability of a solution of a PDE.
The Laguerre iteration in solving the symmetric tridiagonal eigenproblem, revisited
T. Y. Li,Zhonggang Zeng +1 more
TL;DR: This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection, and Cuppen’s divide-and-conquer method and is fully parallel and competitive in speed with the most efficient QR algorithm in serial mode.
36
A Serial Implementation of Cuppen''s Divide and Conquer Algorithm
Jeffery D. Rutter
- 01 Feb 1991
TL;DR: This report discusses a serial implementation of Cuppen''s divide and conquer algorithm for computing all eigenvalues and eigenvectors of a real symmetric matrix, which was uniformly the fastest algorithm by a large margin for large tridiagonal eigenproblems.
32
On the Homotopy Method for Perturbed Symmetric Generalized Eigenvalue Problems
TL;DR: The homotopy method is reviewed and some algorithmic issues such as step size estimation and grouping clustered eigenvalues and grouping eigenvectors are discussed, based on perturbation theory.
24
A Homotopy Method for Finding All Solutions of a Multiparameter Eigenvalue Problem
TL;DR: This paper proposes applying homotopy methods to solve the general multiparameter eigenvalue problems and shows that the proposed method is more efficient for coefficient matrices of large order, and has some adva...
14
References
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TL;DR: In this paper, a pipelined version of EISPACK is presented for finding few or all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix, which consists of isolation, extraction-inverse iteration, and partial orthogonalization.
Solving the symmetric tridiagonal eigenvalues problem on the hypercube
TL;DR: This paper describes implementations of Cuppen's method, bisection, and multisection for the computation of all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix on a distributed-memory hypercube multiprocessor.
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