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Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach

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Summary

The Symmetric Tridiagonal Eigenproblem has been the topic of some recent work. Many methods have been advanced for the computation of the eigenvalues of such a matrix. In this paper, we present a divide-and-conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial. The problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. Furthermore, this set of recursions can be implemented on a regulartree structure. If the concurrency exhibited by this algorithm is exploited, it can be shown that thetime for computation of all the eigenvalues becomesO(nlogn) instead ofO(n 2) as is the case for the approach where the order is increased by only one at every step. We address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation.

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Krishnakumar, A.S., Morf, M. Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach. Numer. Math. 48, 349–368 (1986). https://doi.org/10.1007/BF01389480

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  • DOI: https://doi.org/10.1007/BF01389480

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