Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection

Barth, Wilhelm; Martin, R. S.; Wilkinson, J. H.

doi:10.1007/bf02162154

Cited by 140 publications

(46 citation statements)

References 4 publications

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“…(5) Here there exists a very easy form to solve the characteristic equation det(B − λI) = 0: limits exist for the eigenvalues and the lesser eigenvalue is calculated by bisection method [7].…”

Section: The Proposed Proceduresmentioning

confidence: 99%

A very fast procedure to calculate the smallest singular value

Fraga

2015

2015 Eighth International Conference on Advances in Pattern Recognition (ICAPR)

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The optimization problem of estimate a vector x such that minimize Ax subject to x = 1, where A is a m×n matrix, is frequently found in computer vision. The solution of this problem is the right singular vector associated to the the smallest singular value. This problem must be solved very fast, for example, in real time applications as augmented reality environments are. It is show in this work that the old procedure to calculate directly the smallest singular value and to use one inverse iteration to calculate its associated singular vector is a faster procedure, compared with the state of the art algorithms to calculate the SVD, with relatively small square matrices.

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Section: The Proposed Proceduresmentioning

confidence: 99%

A very fast procedure to calculate the smallest singular value

Fraga

2015

2015 Eighth International Conference on Advances in Pattern Recognition (ICAPR)

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“…As early as 1962, Wilkinson [20] presented bisection as a method for computing the eigenvalues of a symmetric tridiagonal matrix. In 1967, Barth, Martin, and Wilkinson [1] published an improved algorithm that was resistant to overflow and made more efficient use of the Sturm sequences. Information obtained from the Sturm sequences for one zero was used to speed finding the other zeros.…”

Section: Parallel Bisectionmentioning

confidence: 99%

A parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix

Swarztrauber

1993

Math. Comp.

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Abstract. A parallel algorithm, called polysection, is presented for computing the eigenvalues of a symmetric tridiagonal matrix. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found. The use of finite-precision arithmetic may result in multiple zeros; however, in this case, the intervals coalesce and their number determines exactly the multiplicity of the zero. For an NxN matrix the eigenvalues can be determined in 0(log2 A^ time with N2 processors and O(N) time with N processors. The method is compared with a parallel variant of bisection that requires 0(N2) time on a single processor, O(N) time with A^ processors, and 0(log N) time with A^2 processors.

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“…As pointed out by [18] the tridiagonal problem can be the computational bottleneck for large problems taking nearly 70 ∼80% of the total time to solve the entire dense problem. As a result, numerous methods exist for the numerical computation of the eigenvalues of a real tridiagonal matrix to high accuracy, see, e.g., [2,10,11]. To find eigenvalues of a symmetric tridiagonal matrix typically requires O(N 2 ) operations [8], although fast algorithms exist which require O(N ln N) [6].…”

Section: Introductionmentioning

confidence: 99%

Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm

Yang¹

2020

CMR

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A hybrid method is presented for determining maximal eigenvalue and its eigenvector (called eigenpair) of a large, dense, symmetric matrix. Many problems require finding only a small part of the eigenpairs, and some require only the maximal one. In a series of papers, efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements. The key idea is to explicitly construct effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm. In this paper, we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale, dense, symmetric matrix. Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations. We then handle the cases that prevent us from applying Chen's algorithm directly, e.g., the cases with zero or negative super-or sub-diagonal elements. Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.

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Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection

Cited by 140 publications

References 4 publications

A very fast procedure to calculate the smallest singular value

A very fast procedure to calculate the smallest singular value

A parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix

Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm

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