Newton's method for the common eigenvector problem

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.