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The eigenanalysis of matrices or operators based only on the knowledge of the spectrum may be misleading in the non-normal case. Instead of the spectrum, one may fully characterize the spectral behavior of a non-normal matrix by analyzing its spectral portrait, i.e., the set of its resolvent norm. In this paper, we propose a parallel version of the generalized Davidson method for analyzing and plotting the spectral portrait of large non-normal matrices. We report the performance results obtained on the machine Paragon.