The Hyperbolic Case: Two Real Characteristics

2020 ◽  
pp. 179-190
Author(s):  
Otto D. L. Strack
Keyword(s):  
Hyperbolic Case

scholarly journals On Solving Hyperbolic Trajectory Using New Predictor-Corrector Quadrature Algorithms

2014 ◽  
Vol 11 (1) ◽  
pp. 186-192
Author(s):  
Baghdad Science Journal
Keyword(s):  
Quadrature Formula ◽  
Grid Size ◽  
Hyperbolic Orbit ◽  
Efficient Manner ◽  
Hyperbolic Case ◽  
Eccentric Anomaly ◽  
Kepler’S Equation ◽  
Orbit Equation ◽  
Predictor Corrector ◽  
Better Than

In this Paper, we proposed two new predictor corrector methods for solving Kepler's equation in hyperbolic case using quadrature formula which plays an important and significant rule in the evaluation of the integrals. The two procedures are developed that, in two or three iterations, solve the hyperbolic orbit equation in a very efficient manner, and to an accuracy that proves to be always better than 10-15. The solution is examined with and with grid size , using the first guesses hyperbolic eccentric anomaly is and , where is the eccentricity and is the hyperbolic mean anomaly.

Extended algebraic Riccati equations in the abstract hyperbolic case

2000 ◽  
Vol 40 (1-8) ◽  
pp. 105-129 ◽  
Author(s):  
V. Barbu ◽  
I. Lasiecka ◽  
R. Triggiani
Keyword(s):  
Riccati Equations ◽  
Hyperbolic Case ◽  
Algebraic Riccati Equations

scholarly journals CORDIC-like method for solving Kepler’s equation

2018 ◽  
Vol 619 ◽  
pp. A128 ◽  
Author(s):  
M. Zechmeister
Keyword(s):  
Double Precision ◽  
Kepler's Equation ◽  
Hyperbolic Case ◽  
Eccentric Anomaly ◽  
Root Functions ◽  
Kepler’S Equation ◽  
Transcendental Functions ◽  
Halley's Method ◽  
Run Time ◽  
The Cost

Context. Many algorithms to solve Kepler’s equations require the evaluation of trigonometric or root functions. Aims. We present an algorithm to compute the eccentric anomaly and even its cosine and sine terms without usage of other transcendental functions at run-time. With slight modifications it is also applicable for the hyperbolic case. Methods. Based on the idea of CORDIC, our method requires only additions and multiplications and a short table. The table is independent of eccentricity and can be hardcoded. Its length depends on the desired precision. Results. The code is short. The convergence is linear for all mean anomalies and eccentricities e (including e = 1). As a stand-alone algorithm, single and double precision is obtained with 29 and 55 iterations, respectively. Half or two-thirds of the iterations can be saved in combination with Newton’s or Halley’s method at the cost of one division.

scholarly journals Solving Kepler’s equation with CORDIC double iterations

2020 ◽  
Vol 500 (1) ◽  
pp. 109-117
Author(s):  
M Zechmeister
Keyword(s):  
Correction Factor ◽  
Initial Vector ◽  
Kepler's Equation ◽  
Hyperbolic Case ◽  
Eccentric Anomaly ◽  
Rotation Direction ◽  
Changing Scale ◽  
Kepler’S Equation ◽  
Major Shortcoming

ABSTRACT In previous work, we developed the idea to solve Kepler’s equation with a CORDIC-like algorithm, which does not require any division, but still requires multiplications in each iteration. Here we overcome this major shortcoming and solve Kepler’s equation using only bitshifts, additions and one initial multiplication. We prescale the initial vector with the eccentricity and the scale correction factor. The rotation direction is decided without correction for the changing scale. We find that double CORDIC iterations are self-correcting and compensate for possible wrong rotations in subsequent iterations. The algorithm needs 75 per cent more iterations and delivers the eccentric anomaly and its sine and cosine terms times the eccentricity. The algorithm can also be adopted for the hyperbolic case. The new shift-and-add algorithm brings Kepler’s equation close to hardware and allows it to be solved with cheap and simple hardware components.

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