Abstract: The problem of flow stability is often attributed to the eigenvalue problem of giant asymmetric matrix. Most algorithms for solving the eigenvalue problems of giant asymmetric matrices are evolved from the basic Arnoldi algorithm. Firstly, the basic Arnoldi algorithm was briefly described. Secondly, several variants based on Arnoldi algorithm were briefly described, such as explicitly restarted Arnoldi algorithm, implicitly restarted Arnoldi algorithm and multiple implicitly restarted Arnoldi algorithm. Finally, the Arnoldi algorithm and its variants, combined with spectral displacement technology to solve the problem of flow stability. Through numerical experiments, it is found that the multiple implicitly restarted Arnoldi algorithm combined with spectral displacement technology has the highest efficiency.