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In this paper, we propose a new adaptive algorithm for the generalized symmetric eigenvalue problem, which can extract the principal and minor generalized eigenvectors, as well as their corresponding subspaces, at a low computational cost. This algorithm exploits the idea of reduced rank introduced by Davila et al (2000) which transforms the GED problem into a similar one but of reduced dimension that can easily be solved using conventional means. The proposed method is compared to the RLS algorithm by Yang et al (2006) and shown to outperform it w.r.t. both computational cost and convergence rate.