The simplex method for nonlinear mass determinations

The Gauss-Newton method, and calculating a mass that minimizes the variation of residuals are standard techniques for determining planetary masses, but both may fail under certain circums tances. The Gauss-Newton method, in particular, may diverge, and when it converges may converge to a local, rather than global, minimum of the nonlinear regression problem. The simplex method of nonlinear optimization needs no initial estimate for the solution and can be made to converge to a global minimum. It may also be used with non-least squares criteria, such as the L₁ criterion, for greater robustness. But the simplex method achieves these advantages at a high computational price. To test the method as a tool for dynamical astronomy, over 12,000 observations of Neptune were used to calculate Pluto's mass. From an initial estimate of 1/1, 812,000 the Gauss-Newton method diverged. The simplex method converged to a more satisfactory 1/22,000,000 with a range of 1/47,000,000 to 1/14,000,000 as indicated by the mean error. Because the simplex method is considerably slower than competing methods, it should be reserved for refractory problems that do not yield facil solutions when tackled by other methods.