Abstract: A new algorithm, Argument Algorithm, for solving the rootlocus equations is introduced. The algorithm Compromises numerical iteration and analytic operation of the algebraic equations. It is able to predict and adjust the step of iteration by using 1-3 times derivation of the iterative functions with respect to Laplaec operator, S. Therefore, it is easy to solve the convergence problem by using the algorithm. The technique can apply to systems with or without delay.