The solution of an equation or a set of simultaneous equations by proceeding from an initial approximation to a series of repeated trial solutions, each depending upon the immediately preceding approximation, in such a manner that the discrepancy between the newest estimated solution and the true solution is systematically reduced. Newton's method for determining the roots of an algebraic equation is an example of the method of successive approximations. For partial differential equations, the relaxation method is a widely applied example of the method of successive approximations.