  This work discusses the Improved Matrix Method and Weighted Residual Method
for studying the quasinormal modes (QNMs) of black holes. In the first method,
by utilizing Jordan decomposition, the improved matrix method avoids the
calculation of inverse matrices in the original matrix method. This
significantly saves computational resources and time needed to compute the
coefficient square matrices of the derivatives. In the second method, we
illustrate the effectiveness of the weighted residual method in QNMs
calculations by using the collocation method and Galerkin method as examples.
We compared the results of both methods with the Generalized Horowitz-Hubeny
Method and the WKB approximation. The methods proposed in this paper
demonstrates superior precision. Additionally, we provide five criteria to
exclude extraneous roots from the methods.
