Most algorithms for the least-squares estimation of non-linear parameters have centered about either of two approaches. On the one hand, the model may be expanded as a Taylor series and corrections to the several parameters calculated at each iteration on the assumption of local linearity. On the other hand, various modifications of the method of steepest-descent have been used. Both methods not infrequently run aground, the Taylor series method because of divergence of the successive iterates, the steepest-descent (or gradient) methods because of agonizingly slow convergence after the first few iterations. In this paper a maximum neighborhood method is developed which, in effect, performs an optimum interpolation between the Taylor series method and the gradient method, the interpolation being based upon the maximum neighborhood in which the truncated Taylor series gives an adequate representation of the nonlinear model. The results are extended to the problem of solving a set of nonlinear algebraic equations.