<p>Given the efficient application of Chebyshev polynomial acceleration techniques in standard symmetric and non-symmetric eigenvalue problems as well as generalized symmetric eigenvalue problems, we extend this technique to generalized non-symmetric eigenvalue problems and propose the Chebyshev-Davidson method. By partitioning the spectrum of the corresponding shifted matrix, we construct four Chebyshev polynomial filters at each iteration to accelerate the convergence of desired eigenvectors while suppressing the convergence of undesired eigenvectors. The introduction of multiple Chebyshev polynomial filters does not significantly increase the computational cost. Furthermore, to compute several eigenvalues and corresponding eigenvectors of generalized non-symmetric eigenvalue problems, we propose the block Chebyshev-Davidson method. Numerical experiments are carried out to demonstrate its superior performance and robustness compared to some state-of-the-art iterative methods.</p>