Aitken acceleration before Aitken

Kepler solved his eponymous equation M = E − e sin(E) by finding a fixed point of E = M + e sin(E). So guess a value of E and stick it into the right hand side. Then plug that value into the right hand side again. Kepler said a couple iterations should be enough. And a couple iterations are enough if the eccentricity e is small and you don’t need much accuracy. The rate of convergence is determined by e. Kepler implicitly had in mind small values of e because he wasn’t aware of anything orbiting the sun in a highly elliptical orbit. Here’s an example with eccentricity 0.05, about the eccentricity of the orbits of Jupiter and Saturn. from math import sin M, E, e = 1, 1, 0.05 for _ in range(5): E = M + e*sin(E) residual = M - (E - e*sin(E)) print(residual) The residual after just two iterations is 2.77 × 10−5. If you change e to 0.2, the eccentricity of Mercury’s orbit, it takes three iterations to get comparable accuracy. Mercury has the most eccentric orbit of any object Kepler would…