Comparing the fractal basins of attraction in the Hill problem with oblateness and radiation

Abstract

The basins of convergence, associated with the roots (attractors) of a complex equation, are revealed in the Hill problem with oblateness and radiation, using a large variety of numerical methods. Three cases are investigated, regarding the values of the oblateness and radiation. In all cases, a systematic and thorough scan of the complex plane is performed in order to determine the basins of attraction of the several iterative schemes. The correlations between the attracting domains and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly suggests that the basins of convergence, with the highly fractal basin boundaries, produce extraordinary and beautiful formations on the complex plane.

Appendix: Presentation of the iterative schemes

The sixteen iterative formulae, with order of convergence varying from 2 to 16, are the following:

• Newton-Raphson’s method $(p = 2)$: Newton-Raphson’s optimal method (see e.g., Conte and de Boor 1973) is of second order, for simple roots, and the corresponding iterative scheme is given by

  $$ x_{n+1} = x_{n} - u_{n}, $$

  (14)

  where always $u_{n} = f_{n}/f'_{n}$ with $f_{n} = f(x_{n})$ and similarly for the derivatives $f'_{n} = f'(x_{n})$, $f''_{n} = f''(x _{n})$.

• Halley’s method $(p = 3)$: Halley’s method (Halley 1964) is of third order and the corresponding iterative scheme is given by

  $$ x_{n+1} = x_{n} - \frac{u_{n}}{1 - L_{f} u_{n}}, $$

  (15)

  where $L_{f} = f''_{n}/(2f'_{n})$.

• Chebyshev’s method $(p = 3)$: Chebyshev’s method (Traub 1964) is of third order and the corresponding iterative scheme is given by

  $$ x_{n+1} = x_{n} - \biggl( 1 + \frac{L_{f}}{2} \biggr) u_{n}, $$

  (16)

  where $L_{f} = f_{n} f''_{n}/(f'_{n})^{2}$.

• Super Halley’s method $(p = 4)$: Super Halley’s method (Gutiérrez and Hernández 2001) is of fourth order and the corresponding iterative scheme is given by

  $$ x_{n+1} = x_{n} - \biggl( 1 + \frac{L_{f}}{2 ( 1 - L_{f} ) } \biggr) u _{n}, $$

  (17)

  where $L_{f} = f_{n} f''_{n}/(f'_{n})^{2}$

• Modified super Halley’s method $(p = 4)$: Modified super Halley’s optimal method (Chun and Ham 2008) is of fourth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - \frac{2}{3}u_{n}, \\ &x_{n+1} = x_{n} - \biggl( 1 + \frac{L_{f}}{2 ( 1 - L_{f} ) } \biggr) u _{n}, \end{aligned} \end{aligned}$$

  (18)

  where $L_{f} = \frac{f_{n}}{(f'_{n})^{2}} \frac{f'(y_{n}) - f'_{n}}{y _{n} - x_{n}}$.

• King’s method $(p = 4)$: King’s method (King 1973) is of fourth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &x_{n+1} = x_{n} - \frac{ ( f_{n} ) ^{2} + ( \beta - 1 ) f _{n} f(y_{n}) + \beta ( f(y_{n}) ) ^{2}}{f'_{n} ( f_{n} + ( \beta - 2 ) f(y_{n}) ) }, \end{aligned} \end{aligned}$$

  (19)

  where in our experiments we have used $\beta = - 1/2$.

• Jarratt’s method $(p = 4)$: Jarratt’s method (Jarratt 1966) is of fourth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - \frac{2}{3}u_{n}, \\ &x_{n+1} = x_{n} - \frac{u_{n}}{2} - \frac{u_{n}}{2 ( 1 + \frac{3}{2} ( L_{f} - 1 ) ) }, \end{aligned} \end{aligned}$$

  (20)

  where $L_{f} = f'(y_{n})/f'_{n}$.

• Kung-Traub’s method $(p = 4)$: Kung-Traub’s optimal method (Kung and Traub 1974) is of fourth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &x_{n+1} = y_{n} - \frac{f(y_{n})}{f'_{n} ( 1 - L_{f} ) ^{2}}, \end{aligned} \end{aligned}$$

  (21)

  where $L_{f} = f(y_{n})/f_{n}$.

• Maheshwari’s method $(p = 4)$: Maheshwari’s optimal method (Maheshwar 2009) is of fourth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &x_{n+1} = x_{n} + \frac{1}{f'_{n}} \biggl( \frac{f_{n}^{2}}{f(y_{n}) - f_{n}} - \frac{(f(y_{n}))^{2}}{f_{n}} \biggr) . \end{aligned} \end{aligned}$$

  (22)

• Murakami’s method $(p = 5)$: Murakami’s method (Murakami 1978) is of fifth order and the corresponding iterative scheme is given by

  $$ x_{n+1} = x_{n} - a_{1} u_{n} - a_{2} w_{2}(x_{n}) - a_{3} w_{3}(x _{n}) - \psi (x_{n}), $$

  (23)

  where

  $$\begin{aligned}& \begin{aligned} &w_{2}(x_{n}) = \frac{f_{n}}{f'(x_{n} - u_{n})}, \\ &w_{3}(x_{n}) = \frac{f_{n}}{f'(x_{n} + \beta u_{n} + \gamma w_{2}(x _{n}))}, \\ &\psi (x_{n}) = \frac{f_{n}}{b_{1} f'_{n} + b_{2} f'(x_{n} - u_{n})}. \end{aligned} \end{aligned}$$

  (24)

  In our experiments we have used the values: $a_{1} = 0.3$, $a_{2} = -0.5$, $a_{3} = 2/3$, $\beta = -1/2$, $b_{1} = -15/32$, $b_{2} = 75/32$, and $\gamma = 0$.

• Neta’s method $(p = 6)$: Neta’s method (Neta 1979) is of sixth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &z_{n} = y_{n} - \frac{f(y_{n})}{f'_{n}} \frac{f_{n} + \beta f(y_{n})}{f _{n} + ( \beta - 2 ) f(y_{n})}, \\ &x_{n+1} = z_{n} - \frac{f(z_{n})}{f'_{n}} \frac{f_{n} - f(y_{n})}{f _{n} - 3f(y_{n})}. \end{aligned} \end{aligned}$$

  (25)

  In Chun and Neta (2012) it was proved that $\beta = -1/2$ is the best choice.

• Chun-Neta’s method $(p = 6)$: Chun-Neta’s method (Chun and Neta 2012) is of sixth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &z_{n} = y_{n} - \frac{f(y_{n})}{f'_{n} ( 1 - f(y_{n})/f_{n} ) ^{2}}, \\ &x_{n+1} = z_{n} - \frac{f(z_{n})}{f'_{n} ( 1 - f(y_{n})/f_{n} - f(z _{n})/f_{n} ) ^{2}}. \end{aligned} \end{aligned}$$

  (26)

• Neta-Johnson’s method $(p = 8)$: Neta-Johnson’s method (Neta and Johnson 2008) is of eighth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &h_{n} = x_{n} - \frac{1}{8}u_{n} - \frac{3f_{n}}{8f'(y_{n})}, \\ &z_{n} = x_{n} - \frac{f_{n}}{f'_{n}/6 + f'(y_{n})/6 + 2f'(h_{n})/3}, \\ &x_{n+1} = z_{n} - \frac{f(z_{n})}{f'_{n}} \\ &\phantom{x_{n+1} =\,\,}{}\times \frac{f'_{n} + f'(y_{n}) + a_{2}f'(h_{n})}{(-1 - a_{2})f'_{n} + (3 + a_{2})f'(y_{n}) + a_{2} f'(h _{n})}, \end{aligned} \end{aligned}$$

  (27)

  where in our experiments we have used $a_{2} = -1$.

• Neta-Petkovic’s method $(p = 8)$: Neta-Petkovic’s optimal method (Neta and Petković 2010) is of eighth order and the corresponding iterative scheme is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &z_{n} = x_{n} - \frac{f(y_{n})}{f'_{n} ( 1 - f(y_{n})/f_{n} ) ^{2}}, \\ &x_{n+1} = x_{n} - u_{n} + c_{n} f_{n}^{2} - d_{n} f_{n}^{3}, \end{aligned} \end{aligned}$$

  (28)

  where

  $$\begin{aligned}& \begin{aligned} &d_{n} = \frac{1}{(f(y_{n}) - f_{n})(f(y_{n}) - f(z_{n}))} \\ &\phantom{d_{n} =\,\,}{}\times\biggl( \frac{y _{n} - x_{n}}{f(y_{n}) - f_{n}} - \frac{1}{f'_{n}} \biggr) \\ &\phantom{d_{n} =\,\,}{}- \frac{1}{(f(y_{n}) - f(z_{n}))(f(z_{n}) - f_{n})} \\ &\phantom{d_{n} =\,\,}{}\times \biggl( \frac{z _{n} - x_{n}}{f(z_{n}) - f_{n}} - \frac{1}{f'_{n}} \biggr) , \\ &c_{n} = \frac{1}{f(y_{n}) - f_{n}} \biggl( \frac{y_{n} - x_{n}}{f(y _{n}) - f_{n}} - \frac{1}{f'_{n}} \biggr) \\ &\phantom{c_{n} =\,\,}{} - d_{n} \bigl(f(y_{n}) - f_{n}\bigr). \end{aligned} \end{aligned}$$

  (29)

• Neta 14th order method $(p = 14)$: The iterative scheme of Neta’s 14th order method (Neta 1981) is given by

  $$\begin{aligned}& \begin{aligned} &w_{n} = x_{n} - u_{n}, \\ &z_{n} = w_{n} - \frac{f(w_{n})}{f'_{n}} \frac{f_{n} + b f(w_{n})}{f _{n} + (b - 2)f(w_{n})}, \\ &t_{n} = z_{n} - \frac{f(z_{n})}{f'_{n}} \frac{f_{n} - f(w_{n})}{f _{n} - 3f(w_{n})}, \\ &x_{n+1} = x_{n} - u_{n} + c f_{n}^{2} - d f_{n}^{3} + e f_{n}^{4}, \end{aligned} \end{aligned}$$

  (30)

  where

  $$\begin{aligned}& \begin{aligned} &e = \frac{\frac{\phi_{t} - \phi_{z}}{F_{t} - F_{z}} - \frac{\phi_{w} - \phi_{z}}{F_{w} - F_{z}}}{F_{t} - F_{w}}, \\ &d = \frac{\phi_{t} - \phi_{z}}{F_{t} - F_{z}} - e ( F_{t} + F _{z} ) , \\ &c = \phi_{t} - d F_{t} - e F_{t}^{2}, \end{aligned} \end{aligned}$$

  (31)

  where we use the notations

  $$\begin{aligned}& \begin{aligned} &\delta = \delta_{n} - x_{n}, \\ &F_{\delta } = f(\delta_{n}) - f_{n}, \\ &\phi_{\delta } = \frac{\delta }{F_{\delta }^{2}} - \frac{1}{F_{ \delta }f'_{n}}, \end{aligned} \end{aligned}$$

  (32)

  for $\delta = w, z, t$. In our computations we have used the value $b = 2$.

• Neta 16th order method $(p = 16)$: The iterative scheme of Neta’s 16th order optimal method (Neta 1981) is given by

  $$\begin{aligned}& \begin{aligned} &y_{n} = x_{n} - u_{n}, \\ &z_{n} = y_{n} - \frac{f(y_{n})}{f'_{n}} \frac{f_{n} + \beta f(y_{n})}{f _{n} + (\beta - 2)f(y_{n})}, \\ &t_{n} = x_{n} - u_{n} + c_{n} f_{n}^{2} - d_{n} f_{n}^{3}, \\ &x_{n+1} = x_{n} - u_{n} + c f_{n}^{2} - d f_{n}^{3} + e f_{n}^{4}, \end{aligned} \end{aligned}$$

  (33)

  where $c_{n}$ and $d_{n}$ are given by Eqs. (29), while $c$, $d$, and $e$ are given by Eqs. (31). In our computations we have used the value $\beta = 2$.