A scalable eigenvalue solver for symmetric tridiagonal matrices

Trefftz, Christian; Huang, Chengchang; McKinley, Philip K.; Li, T.-Y.; Zeng, Zhonggang

doi:10.1016/0167-8191(95)00019-k

Cited by 15 publications

(4 citation statements)

References 20 publications

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“…Since symmetric tridiagonal matrices with zero diagonal specify their eigenvalues with high relative accuracy independently of their magnitudes [7] numerical methods can possibly compute these eigenvalues at the same relative accuracy they are determined by the input data. Relatively accurate polynomial zerofinders based on Laguerre's iteration are considered in [4,8] while GR-type matrix methods are devised in [7,9]. Similar results do not hold for a general symmetric tridiagonal whose entries do not determine its eigenvalues to high relative accuracy.…”

Section: Introductionmentioning

confidence: 87%

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Gemignani¹

2016

Computers & Mathematics with Applications

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In this paper we consider the application of polynomial root-finding methods to the\ud solution of the tridiagonal matrix eigenproblem. All considered solvers are based on\ud evaluating the Newton correction. We show that the use of scaled three-term recurrence\ud relations complemented with error free transformations yields some compensated\ud schemes which significantly improve the accuracy of computed results at a modest increase\ud in computational cost. Numerical experiments illustrate that under some restriction on\ud the conditioning the novel iterations can approximate and/or refine the eigenvalues of a\ud tridiagonal matrix with high relative accuracy

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Section: Introductionmentioning

confidence: 87%

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Gemignani¹

2016

Computers & Mathematics with Applications

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“…The algorithm employs the split-merge process, similar to Cuppen's divide-and-conquer strategy, which provides an excellent set of starting values that make the algorithm naturally parallel. Impressive speedups of the parallel version of the algorithm were reported in 15]. For 2, 000 2, 000 random matrices, the algorithm can reach a speedup of 52 on an N-cube with 64 nodes.…”

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confidence: 93%

Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems

Jin

Li³

et al. 1996

SIAM J. Sci. Comput.

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In this article, the quasi-Laguerre iteration is established in the spirit of Laguerre's iteration for solving polynomial f with all real zeros. The new algorithm, which maintains the monotonicity and global convergence of the Laguerre iteration, no longer needs to evaluate f". The ultimate convergence rate is + 1. When applied to approximate the eigenvalues of a symmetric tridiagonal matrix, the algorithm substantially improves the speed of Laguerre's iteration.

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“…A continuación fueron presentados muchos otros resultados para el cálculo de la DVS de una matriz: [Arbenz, 1989], [Fernando y Parlett, 1994], [Gu y Eisenstat, 1995a], , [Tre¤tz et al, 1995], [Lang, 1996], [Demmel et al, 1999b], [Fernando, 1998a], [Fernando, 1998b], [Groß er y Lang, 1999], [Groß er y Lang, 2003], [Drmaµ c y Veselić, 2007a], [Drmaµ c y Veselić, 2007b], [Willems et al, 2006], [Howell et al, 2008] y [Demmel et al, 2009].…”

Section: Un Poco De La Historiaunclassified

“…donde U y V son las matrices ortogonales de la bidiagonalización, existen distintos métodos [Arbenz, 1989], [Gu y Eisenstat, 1995a], [Tre¤tz et al, 1995], [Fernando, 1998a], [Fernando, 1998b], [Groß er y Lang, 2003], [Willems et al, 2006] y entre ellos se van a describir los siguientes:…”

Section: Dvs De La Matriz Bidiagonalunclassified

Algoritmos de Altas Prestaciones para el Cálculo de la Descomposición en Valores Singulares y su Aplicación a la Reducción de Modelos de Sistemas Lineales de Control

Campos¹,

Alberto²

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T O calculate the singular value decomposition (SVD) of a real dense matrix, traditional methods start by reducing the initial matrix to a bidiagonal form, and then, calculate the SVD of the bidiagonal matrix.The process of reducing the initial matrix to bidiagonal form is known as the bidiagonalization method, which generally consists of applying successive Householder transformations from the left and right of the matrix. Since the re ‡ectors are applied on both sides of the matrix, this has a negative impact on the communication overheads of a parallel implementation for distributed memory systems.Ralha and Barlow have proposed two new methods for the reduction of dense matrices to bidiagonal form in which the Householder transformations are applied only on the right side of the matrix. This allows to de…ne all operations in terms of full columns of the matrix under transformation. Therefore, these methods are more attractive for distributed memory systems than the standard vii

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A scalable eigenvalue solver for symmetric tridiagonal matrices

Cited by 15 publications

References 20 publications

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems

Algoritmos de Altas Prestaciones para el Cálculo de la Descomposición en Valores Singulares y su Aplicación a la Reducción de Modelos de Sistemas Lineales de Control

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