We discuss a numerical method to compute the homogeneous solutions of the Teukolsky equation which is the basic equation of the black hole perturbati

We extend our previous results of the 14th post-Newtonian (PN) order expansion of gravitational waves for a test particle in circular orbits around a Schwarzschild black hole to the 22PN order, i.e. $v^{44}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. Comparing our 22PN formula for the energy flux with high precision numerical results, we find that the relative error of the 22PN flux at the innermost stable circular orbit is about $10^{-5}$. We also estimate the phase difference between the 22PN waveforms and numerical waveforms after a two-year inspiral. We find that the dephase is about $10^{-9}$ for $\mu/M=10^{-4}$ and $10^{-2}$ for $\mu/M=10^{-5}$ where $\mu$ is the mass of the compact object and $M$ the mass of the central supermassive black hole. Finally, we construct a hybrid formula of the energy flux by supplementing the 4PN formula of the energy flux for circular and equatorial orbits around a Kerr black hole with all the present 22PN terms for the case of a Schwarzschild black hole. Comparing the hybrid formula with the the full numerical results, we examine the performance of the hybrid formula for the case of Kerr black hole.Comment: 22 pages, additional datafiles are available at this http URL