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In this brief, we consider extracting generalized eigenvectors in parallel for the generalized eigendecomposition problem. The problem is formulated as an optimization problem of minimizing an unconstrained quartic cost function based on the weighted rule. It is shown that the proposed weighted cost function has a unique global minimum, which corresponds to the principal generalized eigenvectors. In order to estimate the principal generalized eigenvector matrix efficiently, we simplify the quartic cost function as a quadric one by making an appropriate approximation, and then derive a fast algorithm for extracting the principal generalized eigenvector in parallel. We also show the application of the proposed algorithm in blind source separation. Numerical simulations are performed, and the results demonstrate the performance of the proposed algorithm.