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NOTE: Gauss's method is a preliminary orbit determination, with emphasis on preliminary. The approximation of the Lagrange coefficients and the limitations of the required observation conditions (i.e., insignificant curvature in the arc between observations, refer to Gronchi [2] for more details) causes inaccuracies.
Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems. [6] The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject. [7]
The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function.
Gauss's method, made famous in his 1801 "recovery" of the first lost minor planet, Ceres, has been subsequently polished. One use is in the determination of asteroid masses via the dynamic method. In this procedure Gauss's method is used twice, both before and after a close interaction between two asteroids.
The generalized Gauss–Newton method is a generalization of the least-squares method originally described by Carl Friedrich Gauss and of Newton's method due to Isaac Newton to the case of constrained nonlinear least-squares problems.
Gauss's algorithm for Determination of the day of the week; Gauss's method for preliminary orbit determination; ... additional terms may apply.
At any step in a Gauss-Seidel iteration, solve the first equation for in terms of , …,; then solve the second equation for in terms of just found and the remaining , …,; and continue to . Then, repeat iterations until convergence is achieved, or break if the divergence in the solutions start to diverge beyond a predefined level.
In a simulation this may be implemented by using small time steps for the simulation, using a fixed number of constraint-solving steps per time step, or solving constraints until they are met by a specific deviation. When approximating the constraints locally to first order, this is the same as the Gauss–Seidel method.