Solving Kepler’s Equation: Numerical Methods for Engineering Education

This study compares numerical methods for the elliptic, parabolic, and hyperbolic forms of Kepler’s equation. Newton-type iterations are combined with initial approximations constructed by bisection and local inverse interpolation using linear, quadratic, ordinary cubic, and piecewise cubic Hermite formulas. The central contribution is a systematic construction of stretched grids guided by the singular behaviour of the inverse mapping as $e\to 1^{？}$ and $e\to 1^{+}$ . Regular, geometrically stretched, and analytically optimised grids are compared for elliptic and hyperbolic cases. Continuation, Wynn’s epsilon algorithm, and homotopy are also examined as alternative means of obtaining initial approximations. Numerical results show that concentrating nodes near the parabolic limit substantially improves local Hermite interpolation and usually leaves only a few final Newton corrections.