We introduce a refinement formulation of the KSOR method and establish the rate of convergence of the introduced formulas. We compare a new formula (RKSOR) with the classical form (KSOR). The efficient performance of the new form is established theoretically and confirmed through a numerical example. The decrease in the required number of iterations for convergence is established through the calculation of the spectral radius of the iteration matrices. The calculations and graphs are performed with the help of the computer algebra software Mathematica.