A new method for obtaining approximate solutions of the hyperbolic Kepler's equation

  title={A new method for obtaining approximate solutions of the hyperbolic Kepler's equation},
  author={Martin E. Avendano and Ver'onica Mart'in-Molina and Jorge Ortigas-Galindo},
We provide an approximate zero S̃(g, L) for the hyperbolic Kepler’s equation S − g arcsinh(S)−L = 0 for g ∈ (0, 1) and L ∈ [0,∞). We prove, by using Smale’s α-theory, that Newton’s method starting at our approximate zero produces a sequence that converges to the actual solution S(g, L) at quadratic speed, i.e. if Sn is the value obtained after n iterations, then |Sn − S| ≤ 0.5 n−1|S̃ − S|. The approximate zero S̃(g, L) is a piecewisedefined function involving several linear expressions and one… 
1 Citations

Solving the Kepler equation with the SDG-code

A new code to solve the Kepler equation for elliptic and hyperbolic orbits has been developed. The motivation of the study is the determination of an appropriate seed to initialize the numerical



A Method Solving Kepler’s Equation for Hyperbolic Case

We developed a method to solve Kepler’s equation for the hyperboliccase. The solution interval is separated into three regions; F ≪ 1, F≈ 1, F ≫ 1. For the region F is large, we transformed the

A Solution of Kepler’s Equation

The present study deals with a traditional physical problem: the solution of the Kepler’s equation for all conics (ellipse, hyperbola or parabola). Solution of the universal Kepler’s equation in

A general algorithm for the solution of Kepler's equation for elliptic orbits

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E−M−e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple

On solving Kepler's equation

This work attacks Kepler's equation with the unified derivation of all known bounds and several starting values, a proof of the optimality of these bounds, a very thorough numerical exploration of a large variety of starting values and solution techniques, and finally the best and simplest starting value/solution algorithm: M + e and Wegstein's secant modification of the method of successive substitutions.

The hyperbolic Kepler equation (and the elliptic equation revisited)

A procedure is developed that, in two iterations, solves the hyperbolic Kepler's equation in a very efficient manner, and to an accuracy that proves to be always better than 10−20 (relative

Procedures for solving Kepler's equation

We review starting formulae and iteration processes for the solution of Kepler's equation, and give details of two complete procedures. The first has been in use for a number of years, but the second


It can be optimized for the work of the point estimate on the Newton iteration, reported by Smale at the 20th Congress of Mathematicians in 1986. It has been proved that iffα(z,f) , for every

Bounds on the solution to Kepler's equation

For Kepler's equation two general linear methods of the bounds determination forE0 root are presented. The methods based on elementary properties of convex functions allow an approach toE0 root

Handbook of mathematical formulas and integrals

The z-Transform Numerical Approximation Short Classified Reference List Solutions of Elliptic, Parabolic and Hyperbolic Equations Qualitative Properties of the Heat and Laplace Equation Index.

Complexity and Real Computation

  • L. Blum
  • Mathematics, Computer Science
    Springer New York
  • 1998
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.