On the computation of eigenvectors of a symmetric tridiagonal matrix: comparison of accuracy improvements of Givens and inverse iteration methods
Résumé
The aim of this paper is the comparison of the recent improvements of two methods to compute eigenvectors of a symmetric tridiagonal matrix once the eigenvalues are computed. The first one is the Givens method which is based on the use of Sturm sequences. This method suffers from a lack of accuracy for the computation of the eigenvector when an approximate value (even a very accurate one) of the eigenvalue is used in the computational process. In [Godunov, S.K. and Antonov, A.G. and Kirilyuk, O.P. and Kostin, V.I., Guaranteed accuracy in numerical linear algebra, Mathematics and its Applications, Kluwer Academic Publishers, 1993] the authors introduce a modification of Givens method to ensure the computation of an accurate eigenvector from a good approximation of the corresponding eigenvalue. The second improvement concerns the inverse iteration method. In [Parlett, B.N. and Dhillon, I.S., Fernando's solution to Wilkinson's problem: An application of double factorization, Linear Algebra Appl., 267:247--279, 1997] the authors present a way to determine the best initial vector to start the iterations. Although the two methods and their improvements seem to be very different from a computational point of view, there exists some striking analogies. For instance, in the two methods we look for an optimal index, we have to minimize a residual, etc. In the paper we briefly present the two methods and investigate the connections between them.
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