A tunable non-paraxial accelerating beam generation method is proposed, which can make the beam propagate along arbitrary convex trajectories without paraxial approximation. The explicit expression between amplitude phase in Fourier space and accelerating propagation trajectory is derived by using the Wigner function and the caustics principle. Moreover, a non-paraxial accelerating array beam with self-focusing properties is generated by designing the initial angular spectra of the Fourier space. The influences of beam number, array radius, and beam parameter size on focus position and autofocus performance are analyzed. The results show that the self-focusing array beam generated by this method breaks through the limit of paraxial approximation, and the beam trajectory and control method are more flexible and efficient.