The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^\sigma$ meeting at $T_X$, where $\sigma$ ranges over a Cantor set.Comment: 20 pages, 8 figures. Some passages rewritten for clarit

Study of errors in the integration of the two-body problem using generalized Sundman's anomalies

[EN] As is well known, the numerical integration of the two body problem with constant step presents problems depending on the type of coordinates chosen. It is usual that errors in Runge-Lenz's vector cause an artificial and secular precession of the periaster although the form remains symplectic, theoretically, even when using symplectic methods. Provided that it is impossible to preserve the exact form and all the constants of the problem using a numerical method, a possible option is to make a change in the variable of integration, enabling the errors in the position of the periaster and in the speed in the apoaster to be minimized for any eccentricity value between 0 and 1. The present work considers this casuistry. We provide the errors in norm infinite, of different quantities such as the Energy, the module of the Angular Moment vector and the components of Runge-Lenz's vector, for a large enough number of orbital revolutions.Lopez Orti, JA.; Marco Castillo, FJ.; Martínez Uso, MJ. (2014). Study of errors in the integration of the two-body problem using generalized Sundman's anomalies. SEMA SIMAI Springer Series. 4:105-112. doi:10.1007/978-3-319-06953-1_11S1051124Brower, D., Clemence, G.M.: Celestial Mechanics. Academic, New York (1965)Brumberg, E.V.: Length of arc as independent argument for highly eccentric orbits. Celest. Mech. 53, 323–328 (1992)Fehlberg, E., Marsall, G.C.: Classical fifth, sixth, seventh and eighth Runge–Kutta formulas with stepsize control. Technical report, NASA, R-287 (1968)Ferrándiz, J.M., Ferrer, S., Sein-Echaluce, M.L.: Generalized elliptic anomalies. Celest. Mech. 40, 315–328 (1987)Gragg, W.B.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. SIAM J. Numer. Anal. 2, 384–403 (1965)Janin, G.: Accurate computation of highly eccentric satellite orbits. Celest. Mech. 10, 451–467 (1974)Janin, G., Bond, V.R.: The elliptic anomaly. Technical memorandum, NASA, n. 58228 (1980)Levallois, J.J., Kovalevsky, J.: Géodésie Générale, vol. 4. Eyrolles, Paris (1971)López, J.A., Agost, V., Barreda, M.: A note on the use of the generalized Sundman transformations as temporal variables in celestial mechanics. Int. J. Comput. Math. 89, 433–442 (2012)López, J.A., Marco, F.J., Martínez, M.J.: A study about the integration of the elliptical orbital motion based on a special one-parametric family of anomalies. Abstr. Appl. Anal. 2014, ID 162060, 1–11 (2014)Nacozy, P.: The intermediate anomaly. Celest. Mech. 16, 309–313 (1977)Sundman, K.: Memoire sur le probleme des trois corps. Acta Math. 36, 105–179 (1912)Tisserand, F.F.: Traité de Mecanique Celeste. Gauthier-Villars, Paris (1896)Velez, C.E., Hilinski, S.: Time transformation and Cowell’s method. Celest. Mech. 17, 83–99 (1978