August  2020, 25(8): 3087-3109. doi: 10.3934/dcdsb.2020052

Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders

Department of Applied Mathematics, Dankook University, Cheonan, Korea 330-714

* Corresponding author

Received  April 2019 Revised  October 2019 Published  August 2020 Early access  February 2020

Fund Project: The first author (Y.H. Geum) is supported by research grant NRF-2018R1D1A1B07047715 from National Research Foundation of Korea

An analysis based on an elementary theory of plane curves is presented to locate bifurcation points from a main component in the parameter space of a family of optimal fourth-order multiple-root finders. We explore the basic dynamics of the iterative multiple-root finders under the Möbius conjugacy map on the Riemann sphere. A linear stability theory on local bifurcations is developed from the viewpoint of an arbitrarily small perturbation about the fixed point of the iterative map with a control parameter. Invariant conjugacy properties are established for the fixed point and its multiplier. The parameter spaces and dynamical planes are investigated to analyze the underlying dynamics behind the iterative map. Numerical experiments support the theory of locating bifurcation points of satellite and primitive components in the parameter space.

Citation: Young Hee Geum, Young Ik Kim. Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3087-3109. doi: 10.3934/dcdsb.2020052
References:
[1]

L. V. Ahlfors, Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.

[2]

J. Ainsworth, M. Dawson, J. Pianta and J. Warwick, The Farey Sequence, 2003. Available from: http://www.maths.ed.ac.uk/aar/fareyproject.pdf.

[3]

S. AmatS. Busquier and S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Sci. Ser. A Math. Sci. (N.S.), 10 (2004), 3-35. 

[4]

I. K. Argyros and Á. A. Magreñán, On the convergence of an optimal fourth-order family of methods and its dynamics, Appl. Math. Comput., 252 (2015), 336-346.  doi: 10.1016/j.amc.2014.11.074.

[5]

A. F. Beardon, Iteration of Rational Functions. Complex Analytic Dynamical Systems, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.

[6]

R. BehlA. CorderoS. Motsa and J. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics, Appl. Math. Comput., 265 (2015), 520-532.  doi: 10.1016/j.amc.2015.05.004.

[7]

R. BehlA. CorderoS. MotsaJ. Torregrosa and V. Kanwar, An optimal fourth-order family of methods for multiple roots and its dynamics, Numer. Algorithms, 71 (2016), 775-796.  doi: 10.1007/s11075-015-0023-5.

[8]

R. BehlA. CorderoS. Motsa and J. Torregrosa, Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations, Nonlinear Dynam., 91 (2018), 81-112.  doi: 10.1007/s11071-017-3858-6.

[9]

P. Blanchard, The Dynamics of Newton's Method, Proc. Sympos. Appl. Math., 49, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 1994 doi: 10.1090/psapm/049/1315536.

[10]

B. CamposA. CorderoJ. R. Torregrosa and P. Vindel, Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family, Numer. Algorithms, 73 (2016), 141-156.  doi: 10.1007/s11075-015-0089-0.

[11]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.

[12]

F. ChicharroA. Cordero and J. R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods, Scientific World Journal, 2013 (2013), 1-11.  doi: 10.1155/2013/780153.

[13]

C. Chun, B. Neta and S. Kim, On Jarratt's family of optimal fourth-order iterative methods and their dynamics, Fractals, 22 (2014), 16pp. doi: 10.1142/S0218348X14500133.

[14]

A. CorderoJ. García-MaimóJ. R. TorregrosaM. P. Vassileva and P. Vindel, Chaos in King's iterative family, Appl. Math. Lett., 26 (2013), 842-848.  doi: 10.1016/j.aml.2013.03.012.

[15]

R. L. Devaney, Complex dynamical systems: The mathematics behind the Mandelbrot and Julia sets, Proceedings of Symposia in Applied Mathematics, 49, American Mathematical Society, 1994, 1–29.

[16]

M. García-OlívoJ. M. Gutíerrez and Á. A. Magreñán, A complex dynamical approach of Chebyshev's method, SeMA J., 71 (2015), 57-68.  doi: 10.1007/s40324-015-0046-9.

[17]

Y. H. Geum and Y. I. Kim, A two-parameter family of fourth-order iterative methods with optimal convergence for multiple zeros, J. Appl. Math., 2013 (2013), 1-7.  doi: 10.1155/2013/369067.

[18]

Y. H. GeumY. I. Kim and B. Neta, Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points, J. Comput. Appl. Math., 333 (2018), 131-156.  doi: 10.1016/j.cam.2017.10.033.

[19]

Y. H. GeumY. I. Kim and Á. A. Magreñán, A biparametric extension of King's fourth-order methods and their dynamics, Appl. Math. Comput., 282 (2016), 254-275.  doi: 10.1016/j.amc.2016.02.020.

[20]

Y. H. GeumY. I. Kim and Á. A. Magreñán, A study of dynamics via Mobius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions, J. Comput. Appl. Math., 344 (2018), 608-623.  doi: 10.1016/j.cam.2018.06.006.

[21]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[22]

D. Gulick, Encounters with Chaos, McGraw-Hill Inc., 1992.

[23] A. V. Holden, Chaos, Princeton University Press, Princeton, New Jersey, 1986. 
[24]

H. T. Kung and J. F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach., 21 (1974), 643-651.  doi: 10.1145/321850.321860.

[25]

Á. A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput., 233 (2014), 29-38.  doi: 10.1016/j.amc.2014.01.037.

[26]

A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley & Sons, New York, 2008. doi: 10.1002/9783527617548.

[27]

B. NetaM. Scott and C. Chun, Basin attractors for various methods for multiple roots, Appl. Math. Comput., 218 (2012), 5043-5066.  doi: 10.1016/j.amc.2011.10.071.

[28]

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719468.

[29]

H. Peitgen and P. Richter, The Beauty of Fractals. Images of Complex Dynamical Systems, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61717-1.

[30]

X. WangT. Zhang and Y. Qin, Efficient two-step derivative-free iterative methods with memory and their dynamics, Int. J. Comput. Math., 93 (2016), 1423-1446.  doi: 10.1080/00207160.2015.1056168.

[31]

S. Wolfram, The Mathematica Book, Wolfram Media, Inc., Champaign, 2003.

[32]

X. ZhouX. Chen and Y. Song, Families of third and fourth order methods for multiple roots of nonlinear equations, Appl. Math. Comput., 219 (2013), 6030-6038.  doi: 10.1016/j.amc.2012.12.041.

show all references

References:
[1]

L. V. Ahlfors, Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.

[2]

J. Ainsworth, M. Dawson, J. Pianta and J. Warwick, The Farey Sequence, 2003. Available from: http://www.maths.ed.ac.uk/aar/fareyproject.pdf.

[3]

S. AmatS. Busquier and S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Sci. Ser. A Math. Sci. (N.S.), 10 (2004), 3-35. 

[4]

I. K. Argyros and Á. A. Magreñán, On the convergence of an optimal fourth-order family of methods and its dynamics, Appl. Math. Comput., 252 (2015), 336-346.  doi: 10.1016/j.amc.2014.11.074.

[5]

A. F. Beardon, Iteration of Rational Functions. Complex Analytic Dynamical Systems, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.

[6]

R. BehlA. CorderoS. Motsa and J. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics, Appl. Math. Comput., 265 (2015), 520-532.  doi: 10.1016/j.amc.2015.05.004.

[7]

R. BehlA. CorderoS. MotsaJ. Torregrosa and V. Kanwar, An optimal fourth-order family of methods for multiple roots and its dynamics, Numer. Algorithms, 71 (2016), 775-796.  doi: 10.1007/s11075-015-0023-5.

[8]

R. BehlA. CorderoS. Motsa and J. Torregrosa, Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations, Nonlinear Dynam., 91 (2018), 81-112.  doi: 10.1007/s11071-017-3858-6.

[9]

P. Blanchard, The Dynamics of Newton's Method, Proc. Sympos. Appl. Math., 49, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 1994 doi: 10.1090/psapm/049/1315536.

[10]

B. CamposA. CorderoJ. R. Torregrosa and P. Vindel, Orbits of period two in the family of a multipoint variant of Chebyshev-Halley family, Numer. Algorithms, 73 (2016), 141-156.  doi: 10.1007/s11075-015-0089-0.

[11]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.

[12]

F. ChicharroA. Cordero and J. R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods, Scientific World Journal, 2013 (2013), 1-11.  doi: 10.1155/2013/780153.

[13]

C. Chun, B. Neta and S. Kim, On Jarratt's family of optimal fourth-order iterative methods and their dynamics, Fractals, 22 (2014), 16pp. doi: 10.1142/S0218348X14500133.

[14]

A. CorderoJ. García-MaimóJ. R. TorregrosaM. P. Vassileva and P. Vindel, Chaos in King's iterative family, Appl. Math. Lett., 26 (2013), 842-848.  doi: 10.1016/j.aml.2013.03.012.

[15]

R. L. Devaney, Complex dynamical systems: The mathematics behind the Mandelbrot and Julia sets, Proceedings of Symposia in Applied Mathematics, 49, American Mathematical Society, 1994, 1–29.

[16]

M. García-OlívoJ. M. Gutíerrez and Á. A. Magreñán, A complex dynamical approach of Chebyshev's method, SeMA J., 71 (2015), 57-68.  doi: 10.1007/s40324-015-0046-9.

[17]

Y. H. Geum and Y. I. Kim, A two-parameter family of fourth-order iterative methods with optimal convergence for multiple zeros, J. Appl. Math., 2013 (2013), 1-7.  doi: 10.1155/2013/369067.

[18]

Y. H. GeumY. I. Kim and B. Neta, Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points, J. Comput. Appl. Math., 333 (2018), 131-156.  doi: 10.1016/j.cam.2017.10.033.

[19]

Y. H. GeumY. I. Kim and Á. A. Magreñán, A biparametric extension of King's fourth-order methods and their dynamics, Appl. Math. Comput., 282 (2016), 254-275.  doi: 10.1016/j.amc.2016.02.020.

[20]

Y. H. GeumY. I. Kim and Á. A. Magreñán, A study of dynamics via Mobius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions, J. Comput. Appl. Math., 344 (2018), 608-623.  doi: 10.1016/j.cam.2018.06.006.

[21]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[22]

D. Gulick, Encounters with Chaos, McGraw-Hill Inc., 1992.

[23] A. V. Holden, Chaos, Princeton University Press, Princeton, New Jersey, 1986. 
[24]

H. T. Kung and J. F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach., 21 (1974), 643-651.  doi: 10.1145/321850.321860.

[25]

Á. A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput., 233 (2014), 29-38.  doi: 10.1016/j.amc.2014.01.037.

[26]

A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley & Sons, New York, 2008. doi: 10.1002/9783527617548.

[27]

B. NetaM. Scott and C. Chun, Basin attractors for various methods for multiple roots, Appl. Math. Comput., 218 (2012), 5043-5066.  doi: 10.1016/j.amc.2011.10.071.

[28]

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719468.

[29]

H. Peitgen and P. Richter, The Beauty of Fractals. Images of Complex Dynamical Systems, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61717-1.

[30]

X. WangT. Zhang and Y. Qin, Efficient two-step derivative-free iterative methods with memory and their dynamics, Int. J. Comput. Math., 93 (2016), 1423-1446.  doi: 10.1080/00207160.2015.1056168.

[31]

S. Wolfram, The Mathematica Book, Wolfram Media, Inc., Champaign, 2003.

[32]

X. ZhouX. Chen and Y. Song, Families of third and fourth order methods for multiple roots of nonlinear equations, Appl. Math. Comput., 219 (2013), 6030-6038.  doi: 10.1016/j.amc.2012.12.041.

Figure 1.  Bifurcations on the stability unit circle
Figure 2.  Stability circle $ \boldsymbol S $ for strange fixed point $ z = 1 $
Figure 3.  Stability surfaces of strange fixed points $ \xi = 1 $
Figure 4.  Stability surfaces of the strange fixed points $ \xi_j, 1 \le j \le 2 $ from the roots of $ T(\xi;\lambda) $
Figure 6.  Parameter space $ \mathcal{P} $, red and yellow main components $ \mathcal{H}_1 $
Figure 5.  Color chart defined in Table 1
Figure 7.  Bifurcation points $ {\lambda}_t $ of $ t $-periodic components in $ \mathcal{P} $
Figure 8.  Dynamical planes for various values of $ {\lambda} $-parameters
Figure 9.  Typical geometries for primitive and satellite components
Table 1.  Coloring scheme for a $ q $-periodic orbit with $ q \in \mathbb{N}\cup \{0 \} $
$ q $ $ C_q $
1 $ C_1 (\rm{fixed\; point}\; \infty) $ magenta $ C_1(\rm{fixed \; point} \; 0) $ cyan $ C_1(\rm{fixed \; point} \; 1) $ yellow $ C_{1}(\rm{other\; strange\; fixed\; point}) $ red
$ C_{2} $ $ C_{3} $ $ C_{4} $ $ C_{5} $ $ C_{6} $ $ C_{7} $ $ C_{8} $ $ C_{9} $
orange light green brown blue green dark yellow antiquewhite light pink
$ 2\le q\le 80 $ $ C_{10} $ $ C_{11} $ $ C_{12} $ $ C_{13} $ $ C_{14} $ $ C_{15} $ $ C_{16} $ $ C_{17} $
khaki melon thistle lavender turquoise plum orchid medium orchid
$ C_{18} $ $ C_{19} $ $ C_{20} $ $ C_{21} $ $ C_{22} $ $ C_{23} $ $ C_{24} $ $ C_{25} $
blue violet dark orchid purple powder blue sky blue deep sky blue dodger blue royal blue
$ C_{26} $ $ C_{27} $ $ C_{28} $ $ C_{29} $ $ C_{30} $ $ C_{31} $ $ C_{32} $ $ C_{33} $
medium spring green apple green medium sea green forest green dark blue olive drab bisque moccasin
$ C_{34} $ $ C_{35} $ $ C_{36} $ $ C_{37} $ $ C_{38} $ $ C_{39} $ $ C_{40} $ $ C_{41} $
light salmon salmon light coral Indian red dark red peach puff fire brick sandy brown
$ C_{42} $ $ C_{43} $ $ C_{44} $ $ C_{45} $ $ C_{46} $ $ C_{47} $ $ C_{48} $ $ C_{49} $
wheat tomato orange red chocolate pink pale violet red deep pink violet red
$ C_{50} $ $ C_{51} $ $ C_{52} $ $ C_{53} $ $ C_{54} $ $ C_{55} $ $ C_{56} $ $ C_{57} $
gainsboro light gray dark gray gray charteruse electric indigo electric lime lime
$ C_{58} $ $ C_{59} $ $ C_{60} $ $ C_{61} $ $ C_{62} $ $ C_{63} $ $ C_{64} $ $ C_{65} $
silver teal pale turquoise rosy brown honeydew lemon chiffon misty rose mintcream
$ C_{66} $ $ C_{67} $ $ C_{68} $ $ C_{69} $ $ C_{70} $ $ C_{71} $ $ C_{72} $ $ C_{73} $
gold crimson light crimson lavenderblush slateblue light cyan coral light blue
$ C_{74} $ $ C_{75} $ $ C_{76} $ $ C_{77} $ $ C_{78} $ $ C_{79} $ $ C_{80} $
aquamarine light yellow peru violet papayawhip dark orange sea green
$ \stackrel{q = 0^\ast \mbox{ or}} {q> 80} $ black
∗: q = 0 implies a non-periodic but bounded orbit. These 84 colors are explicitly illustrated in Figure 5.
$ q $ $ C_q $
1 $ C_1 (\rm{fixed\; point}\; \infty) $ magenta $ C_1(\rm{fixed \; point} \; 0) $ cyan $ C_1(\rm{fixed \; point} \; 1) $ yellow $ C_{1}(\rm{other\; strange\; fixed\; point}) $ red
$ C_{2} $ $ C_{3} $ $ C_{4} $ $ C_{5} $ $ C_{6} $ $ C_{7} $ $ C_{8} $ $ C_{9} $
orange light green brown blue green dark yellow antiquewhite light pink
$ 2\le q\le 80 $ $ C_{10} $ $ C_{11} $ $ C_{12} $ $ C_{13} $ $ C_{14} $ $ C_{15} $ $ C_{16} $ $ C_{17} $
khaki melon thistle lavender turquoise plum orchid medium orchid
$ C_{18} $ $ C_{19} $ $ C_{20} $ $ C_{21} $ $ C_{22} $ $ C_{23} $ $ C_{24} $ $ C_{25} $
blue violet dark orchid purple powder blue sky blue deep sky blue dodger blue royal blue
$ C_{26} $ $ C_{27} $ $ C_{28} $ $ C_{29} $ $ C_{30} $ $ C_{31} $ $ C_{32} $ $ C_{33} $
medium spring green apple green medium sea green forest green dark blue olive drab bisque moccasin
$ C_{34} $ $ C_{35} $ $ C_{36} $ $ C_{37} $ $ C_{38} $ $ C_{39} $ $ C_{40} $ $ C_{41} $
light salmon salmon light coral Indian red dark red peach puff fire brick sandy brown
$ C_{42} $ $ C_{43} $ $ C_{44} $ $ C_{45} $ $ C_{46} $ $ C_{47} $ $ C_{48} $ $ C_{49} $
wheat tomato orange red chocolate pink pale violet red deep pink violet red
$ C_{50} $ $ C_{51} $ $ C_{52} $ $ C_{53} $ $ C_{54} $ $ C_{55} $ $ C_{56} $ $ C_{57} $
gainsboro light gray dark gray gray charteruse electric indigo electric lime lime
$ C_{58} $ $ C_{59} $ $ C_{60} $ $ C_{61} $ $ C_{62} $ $ C_{63} $ $ C_{64} $ $ C_{65} $
silver teal pale turquoise rosy brown honeydew lemon chiffon misty rose mintcream
$ C_{66} $ $ C_{67} $ $ C_{68} $ $ C_{69} $ $ C_{70} $ $ C_{71} $ $ C_{72} $ $ C_{73} $
gold crimson light crimson lavenderblush slateblue light cyan coral light blue
$ C_{74} $ $ C_{75} $ $ C_{76} $ $ C_{77} $ $ C_{78} $ $ C_{79} $ $ C_{80} $
aquamarine light yellow peru violet papayawhip dark orange sea green
$ \stackrel{q = 0^\ast \mbox{ or}} {q> 80} $ black
∗: q = 0 implies a non-periodic but bounded orbit. These 84 colors are explicitly illustrated in Figure 5.
Table 2.  Typical $ \ell/q $–bifurcation points $ {\lambda} $ for $ 1\le q \le 10 $ and $ 0 \le \ell \le 9 $
$ \ell $
$ q $ 0 1 2 3 4 5 6 7 8 9
$ 1 $ -4.4
$ 2 $ -3.81818
$ 3 $ $ \begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix}^\ast $ $ \begin{pmatrix}-3.85567\\0.14285 \end{pmatrix} $
$ 4 $ $ \begin{pmatrix}-3.91781 \\- 0.219178\end{pmatrix} $ $ \begin{pmatrix}-3.91781\\0.219178 \end{pmatrix} $
$ 5 $ $ \begin{pmatrix}-3.98185 \\- 0.261606 \end{pmatrix} $ $ \begin{pmatrix}-3.8306 \\- 0.0840949 \end{pmatrix} $ $ \begin{pmatrix}-3.8306\\0.0840949 \end{pmatrix} $ $ \begin{pmatrix}-3.98185\\0.261606 \end{pmatrix} $
$ 6 $ $ \begin{pmatrix}-4.04082 \\- 0.282784 \end{pmatrix} $ $ \begin{pmatrix}-4.04082\\0.282784 \end{pmatrix} $
$ 7 $ $ \begin{pmatrix}-4.09231 \\- 0.290424 \end{pmatrix} $ $ \begin{pmatrix}-3.88575 \\- 0.186408\end{pmatrix} $ $ \begin{pmatrix}-3.82438 \\- 0.0597191 \end{pmatrix} $ $ \begin{pmatrix}-3.82438\\0.0597191 \end{pmatrix} $ $ \begin{pmatrix}-3.88575\\0.186408 \end{pmatrix} $ $ \begin{pmatrix}-4.09231\\0.290424 \end{pmatrix} $
$ 8 $ $ \begin{pmatrix}-4.13604 \\- 0.289658\end{pmatrix} $ $ \begin{pmatrix}-3.8381 \\- 0.105794 \end{pmatrix} $ $ \begin{pmatrix}-3.8381\\0.105794 \end{pmatrix} $ $ \begin{pmatrix}-4.13604\\0.289658 \end{pmatrix} $
$ 9 $ $ \begin{pmatrix}-4.17269 \\- 0.283871 \end{pmatrix} $ $ \begin{pmatrix}-3.95018 \\- 0.24367 \end{pmatrix} $ $ \begin{pmatrix}-3.8219 \\- 0.0463343 \end{pmatrix} $ $ \begin{pmatrix}-3.8219\\0.0463343 \end{pmatrix} $ $ \begin{pmatrix}-3.95018\\0.24367\end{pmatrix} $ $ \begin{pmatrix}-4.17269\\0.283871 \end{pmatrix} $
$ 10 $ $ \begin{pmatrix}-4.20324 \\- 0.275251 \end{pmatrix} $ $ \begin{pmatrix}-3.8754 \\- 0.173249\end{pmatrix} $ $ \begin{pmatrix}-3.8754\\0.173249\end{pmatrix} $ $ \begin{pmatrix}-4.20324\\0.275251\end{pmatrix} $
$ {}^\ast $: $ \begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix} \equiv -3.85567- 0.14285 \; i, \; i = \sqrt{-1} $
$ \ell $
$ q $ 0 1 2 3 4 5 6 7 8 9
$ 1 $ -4.4
$ 2 $ -3.81818
$ 3 $ $ \begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix}^\ast $ $ \begin{pmatrix}-3.85567\\0.14285 \end{pmatrix} $
$ 4 $ $ \begin{pmatrix}-3.91781 \\- 0.219178\end{pmatrix} $ $ \begin{pmatrix}-3.91781\\0.219178 \end{pmatrix} $
$ 5 $ $ \begin{pmatrix}-3.98185 \\- 0.261606 \end{pmatrix} $ $ \begin{pmatrix}-3.8306 \\- 0.0840949 \end{pmatrix} $ $ \begin{pmatrix}-3.8306\\0.0840949 \end{pmatrix} $ $ \begin{pmatrix}-3.98185\\0.261606 \end{pmatrix} $
$ 6 $ $ \begin{pmatrix}-4.04082 \\- 0.282784 \end{pmatrix} $ $ \begin{pmatrix}-4.04082\\0.282784 \end{pmatrix} $
$ 7 $ $ \begin{pmatrix}-4.09231 \\- 0.290424 \end{pmatrix} $ $ \begin{pmatrix}-3.88575 \\- 0.186408\end{pmatrix} $ $ \begin{pmatrix}-3.82438 \\- 0.0597191 \end{pmatrix} $ $ \begin{pmatrix}-3.82438\\0.0597191 \end{pmatrix} $ $ \begin{pmatrix}-3.88575\\0.186408 \end{pmatrix} $ $ \begin{pmatrix}-4.09231\\0.290424 \end{pmatrix} $
$ 8 $ $ \begin{pmatrix}-4.13604 \\- 0.289658\end{pmatrix} $ $ \begin{pmatrix}-3.8381 \\- 0.105794 \end{pmatrix} $ $ \begin{pmatrix}-3.8381\\0.105794 \end{pmatrix} $ $ \begin{pmatrix}-4.13604\\0.289658 \end{pmatrix} $
$ 9 $ $ \begin{pmatrix}-4.17269 \\- 0.283871 \end{pmatrix} $ $ \begin{pmatrix}-3.95018 \\- 0.24367 \end{pmatrix} $ $ \begin{pmatrix}-3.8219 \\- 0.0463343 \end{pmatrix} $ $ \begin{pmatrix}-3.8219\\0.0463343 \end{pmatrix} $ $ \begin{pmatrix}-3.95018\\0.24367\end{pmatrix} $ $ \begin{pmatrix}-4.17269\\0.283871 \end{pmatrix} $
$ 10 $ $ \begin{pmatrix}-4.20324 \\- 0.275251 \end{pmatrix} $ $ \begin{pmatrix}-3.8754 \\- 0.173249\end{pmatrix} $ $ \begin{pmatrix}-3.8754\\0.173249\end{pmatrix} $ $ \begin{pmatrix}-4.20324\\0.275251\end{pmatrix} $
$ {}^\ast $: $ \begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix} \equiv -3.85567- 0.14285 \; i, \; i = \sqrt{-1} $
Table 3.  Typical numerical values of $ \{\xi, {\lambda} \} $ for satellite and primitive components with $ q k \le 8 $
$ (q, k) $ Type $ \xi $ $ \lambda $ Fig. No.
$ (2, 1) $ Primitive $ -0.23512621166835 + 1.6020106612492319\; i $ $ -1.951962936703127 -1.7485319734836857\; i $ 7(a)
$ (3, 1) $ Primitive $ 0.454659800907803 + 0\; i $ $ -5.219261654371707 + 0\; i $ 7(b)
$ (4, 1) $ Primitive $ 0.341977833651053 + 0.4814672920477362\; i $ $ -3.564458921821090 + 1.646535885566244\; i $ 7(c)
$ (5, 1) $ Primitive $ 0.757190777481740 + 0.2122872817258981\; i $ $ -4.045956427432157 + 0.346070091333350\; i $ 7(d)
$ (6, 1) $ Primitive $ -1.10304555596122 + 1.0059215339291120\; i $ $ -0.568308288946734 - 0.953277135441219\; i $ 7(e)
$ (7, 1) $ Primitive $ 0.764464537114743 + 0.2097254339881594\; i $ $ -4.018513045186225 + 0.345332811593217\; i $ 7(f)
$ (2, 2) $ Satellite $ 0.851625771743808 + 0.2369760927165886\; i $ $ -3.752943559023020 + 0.074408072396025\; i $ 7(g)
$ (3, 2) $ Satellite $ 0.875562175989576 + 0.2566238834815281\; i $ $ -3.838763107389483 + 0.152045173864673\; i $ 7(h)
$ (4, 2) $ Satellite $ 1.201354999600545 - 0.2912383604676560\; i $ $ -3.921140825708978 + 0.257336038244326\; i $ 7(i)
$ (q, k) $ Type $ \xi $ $ \lambda $ Fig. No.
$ (2, 1) $ Primitive $ -0.23512621166835 + 1.6020106612492319\; i $ $ -1.951962936703127 -1.7485319734836857\; i $ 7(a)
$ (3, 1) $ Primitive $ 0.454659800907803 + 0\; i $ $ -5.219261654371707 + 0\; i $ 7(b)
$ (4, 1) $ Primitive $ 0.341977833651053 + 0.4814672920477362\; i $ $ -3.564458921821090 + 1.646535885566244\; i $ 7(c)
$ (5, 1) $ Primitive $ 0.757190777481740 + 0.2122872817258981\; i $ $ -4.045956427432157 + 0.346070091333350\; i $ 7(d)
$ (6, 1) $ Primitive $ -1.10304555596122 + 1.0059215339291120\; i $ $ -0.568308288946734 - 0.953277135441219\; i $ 7(e)
$ (7, 1) $ Primitive $ 0.764464537114743 + 0.2097254339881594\; i $ $ -4.018513045186225 + 0.345332811593217\; i $ 7(f)
$ (2, 2) $ Satellite $ 0.851625771743808 + 0.2369760927165886\; i $ $ -3.752943559023020 + 0.074408072396025\; i $ 7(g)
$ (3, 2) $ Satellite $ 0.875562175989576 + 0.2566238834815281\; i $ $ -3.838763107389483 + 0.152045173864673\; i $ 7(h)
$ (4, 2) $ Satellite $ 1.201354999600545 - 0.2912383604676560\; i $ $ -3.921140825708978 + 0.257336038244326\; i $ 7(i)
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