Rootfinding for a transcendental equation without a first guess: Polynomialization of Kepler's equation through Chebyshev polynomial expansion of the sine

Abstract

The Kepler equation for the parameters of an elliptical orbit, E − ε sin ( E ) = M , is reduced from a transcendental to a polynomial equation by expanding the sine as a series of Chebyshev polynomials. The single real root is found by applying standard polynomial rootfinders and accepting only the polynomial root that lies on the interval predicted by rigorous theoretical bounds. A complete Matlab implementation is given in full because it requires just seven lines. For a polynomial of degree fifteen, the maximum absolute error over the whole range ε ∈ [ 0 , 1 ] and all M is only 4 × 10 −10 . Other transcendental equations can similarly be reduced to polynomial equations through Chebyshev expansions.