Abstract: In this paper, a new numerical method for the solution of radial Hartree-Fock-Slater equation is proposed. The discretized form of the HFSequation, using second-order or 4th-order finite differences, results in a set of quasi-linear algebraic eigenvalue problems. The coefficient-matrices are dependent on the solutions. After linearizing the coefficient-matrices, we can solve the eigenvalue problems by the Newton's method to obtain the eigenvalues and the corresponding eigenvectors simultaneously. We solve he linear system therein by the double sweep method. Then we can correct the coefficient-matrices by the simple weighting iterate method or by the Newton's method. This numerical method has been programmed for computer. The results of trial calculations of Cu⁺ and Mn²⁺ are consistent with those in published literature. Actual calculation shows that the stability of this numerical method is good.