Controllability Properties and Invariance Pressure for Linear Discrete-Time Systems

For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure of this control set is proved.

Solving Kepler’s equation with CORDIC double iterations

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.

Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space.

CORDIC-like method for solving Kepler’s equation

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.