The Conditions for the Convergence of Power Scaled Matrices and Applications
Xuzhou Chen, Robert E. Hartwig
DOI: 10.4236/ajcm.2011.12007   PDF    HTML     4,685 Downloads   9,368 Views  


For an invertible diagonal matrix D , the convergence of the power scaled matrix sequence D-NAN is investigated. As a special case, necessary and sufficient conditions are given for the convergence of D-NTN , where T is triangular. These conditions involve both the spectrum as well as the diagraph of the matrix .The results are then used to privide a new proof for the convergence of subspace iteration.

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X. Chen and R. Hartwig, "The Conditions for the Convergence of Power Scaled Matrices and Applications," American Journal of Computational Mathematics, Vol. 1 No. 2, 2011, pp. 63-71. doi: 10.4236/ajcm.2011.12007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] F. L. Bauer, “Das Verfahren der Treppeniteration und verwandte Verfahren zur L?sung Algebraischer Eigen- wertprobleme,” Zeitschrift für Angewandte Mathe- matik und Physik, Vol. 8, No. 3, 1957, pp. 214-235. doi:10.1007/BF01600502
[2] A. Ben-Israel, “A Volume Associated with Matrices,” Linear Algebra and Its Applications, Vol. 167, No. 1, pp. 87-111, 1992. doi:10.1016/0024-3795(92)90340-G
[3] X. Chen and R. E. Hartwig, “On Simultaneous Iteration for Computing the Schur Vectors of Matrices,” Pro- ceedings of the 5th SIAM Conference on Applied Linear Algebra, Snowbird, June 13-19, 1994, pp. 290-294.
[4] X. Chen and R. E. Hartwig, “On the Convergence of Power Scaled Cesaro Sums,” Linear Algebra and Its Applications, Vol. 267, pp. 335-358, 1997. doi:10.1016/S0024-3795(97)80056-0
[5] X. Chen and R. E. Hartwig, “The Semi-iterative Method Applied to the Hyperpower Iteration,” Numerical Linear Algebra with Applications, Vol. 12, No. 9, pp. 895-910, 2005. doi:10.1002/nla.429
[6] X. Chen and R. E. Hartwig, “The Picard Iteration and Its Application,” Linear and Multi-linear Algebra, Vol. 54, No. 5, 2006, pp. 329-341. doi:10.1080/03081080500209703
[7] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, Cambridge, 1985.
[8] R. A. Horn and C. R. Johnson, “Topics in Matrix Ana- lysis,” Cambridge University Press, Cambridge, 1991.
[9] A. S. Householder, “The Theory of Matrices in Nu- merical Analysis,” Dover, New York, 1964.
[10] B. N. Parlett and W. G. Poole, Jr., “A Geometric Theory for the QR, LU, and Power Iterations,” SIAM Journal on Numerical Analysis, Vol. 10, No. 2, 1973, pp. 389-412. doi:10.1137/0710035
[11] H. Rutishauser, “Simultaneous Iteration Method for Symmetric Matrices,” Numerische Mathe-matik, Vol. 16, No. 3, 1970, pp. 205-223. doi:10.1007/BF02219773
[12] Y. Saad, “Numerical Methods for Large Eigenvalue Problems,” Manchester University Press, Manchester, 1992.
[13] G. W. Stewart, “Methods of Simultaneous Iteration for Calculating Eigenvectors of Matrices” In: John J. H. Miller, Eds., Topics in Numerical Analysis II, Academic Press, New York, 1975, pp. 185-169.
[14] G. R. Wang, Y. Wei and S. Qiao, “Generalized Inverses: Theory and Computations,” Science Press, Beijing/New York, 2004.
[15] D. S. Watkins, “Understanding the QR Algorithm,” SIAM Review, Vol. 24, No. 4, 1982, pp. 427-440. doi:10.1137/1024100
[16] D. S. Watkins, “Some Perspectives on the Eigenvalue Problem,” SIAM Review, Vol. 35, No. 3, 1993, pp. 430-470, doi:10.1137/1035090
[17] J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Oxford University Press (Clarendon), London and New York, 1964.

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