A procedure solving the extended Kepler's equation for the hyperbolic case

The bisection method to localize the solution of a nonlinear equation [Fukushima (1996, AJ, 112, 2858)] was extended to handle a long sequence of bisections in a concise manner. This was done by means of the addition theorem of transcendental functions appearing in the equation. The localizer extended was combined with a variation of Newton's method where the functions an evaluated by their Taylor series expansions. As its application, we developed a procedure solving an extended form of Kepler's equation for the hyperbolic case. Our procedure is robust and fast. It finds sufficiently precise solutions even when Danby's starter [1988, Fundamentals of Celestial Mechanics, 2nd Ed. (Willmann-Bell, Richmond, VA), Section 6.9] fails. For typical cases, it requires less CPU time than that for evaluating the equation itself for an arbitrary argument. (C) 1997 American Astronomical Society.