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In this paper, a new algorithm for Sparse Component Analysis (SCA) or atomic decomposition on over-complete dictionaries is presented. The algorithm is essentially a method for obtaining sufficiently sparse solutions of underdetermined systems of linear equations. The solution obtained by the proposed algorithm is compared with the minimum ℓ¹-norm solution achieved by Linear Programming (LP). It is experimentally shown that the proposed algorithm is about two orders of magnitude faster than the state-of-the-art ℓ¹-magic, while providing the same (or better) accuracy.