A MONOTONIC STARTER FOR SOLVING THE HYPERBOLIC KEPLER EQUATION BY NEWTON'S METHOD

This communication deals with the iterative solution of the sine hyperbolic Kepler's equation (SHK): F-g(S) = S - g arcsinh(S) - L = 0. Since this function is monotonic increasing and convex, any starter value S-0 such that F-g(S-0) > 0, leads to a Newton's sequence S-j monotic decreasing to the exact solution of SKE equation and therefore has some advantages over non monotonic starters. Because of this, we are able to construct a monotonic starter such that minimizes the computational cost and that guarantees super-convergence (q-convergence) by analyzing the error estimates of Newton's iteration. In contrast with other starters in which the quality is assessed by extensive numerical experiments, here we use theoretical tools to reach super-convergence.