Abstract
A simple algorithm for the computation of eigenvalues of real or complex square matrices is proposed. This algorithm is based on an additive decomposition of the matrix. A sufficient condition for convergence is proved. It is also shown that this method has many properties of the QR algorithm : it is invariant for the Hessenberg form, shifts are possible in the case of a null element on the diagonal. Some other interesting experimental properties are shown. Numerical experiments are given showing that most of the time the behavior of this method is not much different from that of the QR method, but sometimes it gives better results, particularly in the case of a bad conditioned real matrix having real eigenvalues.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alt, R.: Un algorithme simple et efficace de calcul de valeurs propres. C. R. Acad Sc. Paris 306(1), 437–440 (1988)
Alt, R., Markov, S.: On the Algebraic Properties of Stochastic Arithmetic. Comparison to Interval Arithmetic. In: Kraemer, W., von Gudenberg, J.W. (eds.) Scientific Computing, Validated Numerics, Interval Methods, pp. 331–341. Kluwer Academic Publishers, Dordrecht (2001)
Alt, R., Lamotte, J.-L., Markov, S.: Numerical Study of Algebraic Solutions to Linear Problems Involving Stochastic Parameters. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2005. LNCS, vol. 3743, pp. 273–280. Springer, Heidelberg (2006)
Bauer, F.L., Fike, C.T.: Norms and exclusion theorems. Numer. Math. 2, 137–141 (1960)
Gregory, R., Kerney, D.: A collection of matrices for testing computational algorithms. Wiley, New-York (1969)
Higham, N.: Algorithm 694: A Collection of Test Matrices in MATLAB. ACM Transactions on Mathematical Software 17(3), 289–305 (1991)
Markov, S., Alt, R.: Stochastic arithmetic: Addition and multiplication by scalars. Applied Num. Math. 50, 475–488 (2004)
Vignes, J., Alt, R.: An Efficient Stochastic Method for Round-Off Error Analysis. In: Miranker, W.L., Toupin, R.A. (eds.) Accurate Scientific Computations. LNCS, vol. 235, pp. 183–205. Springer, Heidelberg (1986)
Vignes, J.: A Stochastic Arithmetic for Reliable Scientific Computation. Math. Comp. in Sim. 35, 233–261 (1993)
Westlake, J.R.: A handbook of Numerical Matrix inversion and solution of linear equations, pp. 136–157. Wiley, Chichester (1968)
Wilkinson, J.: Convergence of the LR, QR and related algorithms. Computer J. 4, 77–84 (1965)
Wilkinson, J.H., Reinsch, C. (eds.): Linear Algebra. Handbook for Automatic Computation vol. II. Springer, New York (1971)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Alt, R. (2007). A Simple and Efficient Algorithm for Eigenvalues Computation. In: Boyanov, T., Dimova, S., Georgiev, K., Nikolov, G. (eds) Numerical Methods and Applications. NMA 2006. Lecture Notes in Computer Science, vol 4310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70942-8_32
Download citation
DOI: https://doi.org/10.1007/978-3-540-70942-8_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70940-4
Online ISBN: 978-3-540-70942-8
eBook Packages: Computer ScienceComputer Science (R0)