Newton's method for the common eigenvector problem

  • Authors:

  • Abdellatif El Ghazi;Said El Hajji;Luc Giraud;Serge Gratton

  • Affiliations:

  • Facultés des Sciences, Université Mohammed V-Agdal, Rabat, Morocco;Facultés des Sciences, Université Mohammed V-Agdal, Rabat, Morocco;CERFACS, 42 av. Gaspard Coriolis, 31057 Toulouse, Cedex, France;CERFACS, 42 av. Gaspard Coriolis, 31057 Toulouse, Cedex, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2008

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Abstract

In El Ghazi et al. [Backward error for the common eigenvector problem, CERFACS Report TR/PA/06/16, Toulouse, France, 2006], we have proved the sensitivity of computing the common eigenvector of two matrices A and B, and we have designed a new approach to solve this problem based on the notion of the backward error. If one of the two matrices (saying A) has n eigenvectors then to find the common eigenvector we have just to write the matrix B in the basis formed by the eigenvectors of A. But if there is eigenvectors with multiplicity 1, the common vector belong to vector space of dimension 1 and such strategy would not help compute it. In this paper we use Newton's method to compute a common eigenvector for two matrices, taking the backward error as a stopping criteria. We mention that no assumptions are made on the matrices A and B.