A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices

We propose a new fast algorithm for computing the spectrum of an N×N symmetric tridiagonal matrix in O(NlnN) operations. Such an algorithm may be combined with any of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying t...

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Bibliographic Details
Published in:Applied and computational harmonic analysis Vol. 34; no. 3; pp. 379 - 414
Main Authors: Coakley, Ed S., Rokhlin, Vladimir
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2013
Subjects:
Algorithms
Computation
Eigenvalues
Fast multipole method
FORTRAN
Mathematical analysis
Matrices
Matrix methods
Spectra
Symmetric eigenproblem
Tridiagonal
Symmetric eigenproblem
Tridiagonal
Fast multipole method
ISSN:1063-5203, 1096-603X
Online Access:Get full text
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Summary:We propose a new fast algorithm for computing the spectrum of an N×N symmetric tridiagonal matrix in O(NlnN) operations. Such an algorithm may be combined with any of the existing methods for the determination of eigenvectors of a symmetric tridiagonal matrix with known eigenvalues. The underlying technique is a divide-and-conquer approach which determines eigenvalues of a larger tridiagonal matrix from those of constituent matrices by the use of relations of their characteristic polynomials. The evaluation of characteristic polynomials is accelerated by the use of a technique known as the fast multipole method. An implementation of the algorithm has been developed in Fortran, providing for a comparison with existing techniques in terms of running time and accuracy. We present numerical results which demonstrate the effectiveness of the method.
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ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2012.06.003