American Institute of Mathematical Sciences

Advanced
December  2019, 6(2): 361-383. doi: 10.3934/jcd.2019018

A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation

Yuto Miyatake 1, , Tai Nakagawa 2, , Tomohiro Sogabe 3, and  Shao-Liang Zhang 3,

1. 

Cybermedia Center, Osaka University, 1-32 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
2. 

Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
3. 

Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

Received  February 2019 Revised  September 2019 Published  November 2019

We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.

Keywords: Fractional Schrödinger equation, structure-preservation, energy-preservation, pseudo-spectral method, Bi-CGSTAB method, preconditioning.
Mathematics Subject Classification: Primary: 65M70, 65F08; Secondary: 65N22.
Citation: Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018
References:
[1]

C. Besse, A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.

[2]

C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy preserving methods for nonlinear Schrödinger equations, (2018).

[3]

E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789.  doi: 10.1016/j.jcp.2012.06.022.

[4]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.

[5]

M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33 (2011), 2318-2340.  doi: 10.1137/100810174.

[6]

M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288.  doi: 10.1016/0021-9991(81)90052-8.

[7]

S. W. Duo and Y. Z. Zhang, Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71 (2016), 2257-2271.  doi: 10.1016/j.camwa.2015.12.042.

[8]

E. Faou, Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2012. doi: 10.4171/100.

[9]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition, Emended and with a preface by Daniel F. Styer. Dover Publications, Inc., Mineola, NY, 2010.

[10] D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011. 
[11]

B. L. Guo, Y. Q. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 468-477.  doi: 10.1016/j.amc.2008.07.003.

[12]

B. L. Guo and Z. H. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.

[13]

X. Y. Guo and M. Y. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 9 pp. doi: 10.1063/1.2235026.

[14]

K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[15]

N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10541.

[16]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.

[17]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[18]

M. Li, X.-M. Gu, C. M. Huang, M. F. Fei and G. Y. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys., 358 (2018), 256-282.  doi: 10.1016/j.jcp.2017.12.044.

[19]

M. Li, C. Huang and W. Ming, A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations, Numer. Algorithms, (2019), 1–26. doi: 10.1007/s11075-019-00672-3.

[20]

M. Li, C. M. Huang and P. D. Wang, Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algorithms, 74 (2017), 499-525.  doi: 10.1007/s11075-016-0160-5.

[21]

S. Longhi, Fractional Schrödinger equation in optics, Opt. Lett., 40 (2015), 1117. doi: 10.1364/OL.40.001117.

[22]

T. Matsuo and D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), 425-447.  doi: 10.1006/jcph.2001.6775.

[23]

Y. Miyatake and T. Matsuo, Conservative finite difference schemes for the Degasperis-Procesi equation, J. Comput. Appl. Math., 236 (2012), 3728-3740.  doi: 10.1016/j.cam.2011.09.004.

[24]

Y. Miyatake, T. Matsuo and D. Furihata, Invariants-preserving integration of the modified Camassa-Holm equation, Jpn. J. Ind. Appl. Math., 28 (2011), 351-381.  doi: 10.1007/s13160-011-0043-z.

[25]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7 pp. doi: 10.1088/1751-8113/41/4/045206.

[26]

T. Sogabe and S.-L. Zhang, A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math., 199 (2007), 297-303.  doi: 10.1016/j.cam.2005.07.032.

[27]

H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), 631-644.  doi: 10.1137/0913035.

[28]

H. A. van der Vorst and J. B. Melissen, A Petrov-Galerkin type method for solving ${A}x = b$, where ${A}$ is symmetric complex, IEEE Trans. Mag., 26 (1990), 706-708. 

[29]

D. L. Wang, A. G. Xiao and W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys., 272 (2014), 644-655.  doi: 10.1016/j.jcp.2014.04.047.

[30]

P. D. Wang and C. M. Huang, A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation, Numer. Algorithms, 69 (2015), 625-641.  doi: 10.1007/s11075-014-9917-x.

[31]

P. D. Wang and C. M. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238-251.  doi: 10.1016/j.jcp.2014.03.037.

[32]

P. D. Wang and C. M. Huang, Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions, Comput. Math. Appl., 71 (2016), 1114-1128.  doi: 10.1016/j.camwa.2016.01.022.

[33]

P. D. Wang and C. M. Huang, Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math., 129 (2018), 137-158.  doi: 10.1016/j.apnum.2018.03.008.

[34]

P. D. Wang, C. M. Huang and L. B. Zhao, Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, J. Comput. Appl. Math., 306 (2016), 231-247.  doi: 10.1016/j.cam.2016.04.017.

[35]

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, Y. P. Zhang and M. Xiao, Propagation dynamics of a light beam in a fractional Schrödinger equation, Phys. Rev. Lett., 115 (2015), 180403. doi: 10.1103/PhysRevLett.115.180403.

show all references

References:
[1]

C. Besse, A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.

[2]

C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy preserving methods for nonlinear Schrödinger equations, (2018).

[3]

E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789.  doi: 10.1016/j.jcp.2012.06.022.

[4]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.

[5]

M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33 (2011), 2318-2340.  doi: 10.1137/100810174.

[6]

M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288.  doi: 10.1016/0021-9991(81)90052-8.

[7]

S. W. Duo and Y. Z. Zhang, Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71 (2016), 2257-2271.  doi: 10.1016/j.camwa.2015.12.042.

[8]

E. Faou, Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2012. doi: 10.4171/100.

[9]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition, Emended and with a preface by Daniel F. Styer. Dover Publications, Inc., Mineola, NY, 2010.

[10] D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011. 
[11]

B. L. Guo, Y. Q. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 468-477.  doi: 10.1016/j.amc.2008.07.003.

[12]

B. L. Guo and Z. H. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.

[13]

X. Y. Guo and M. Y. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 9 pp. doi: 10.1063/1.2235026.

[14]

K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[15]

N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10541.

[16]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.

[17]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[18]

M. Li, X.-M. Gu, C. M. Huang, M. F. Fei and G. Y. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys., 358 (2018), 256-282.  doi: 10.1016/j.jcp.2017.12.044.

[19]

M. Li, C. Huang and W. Ming, A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations, Numer. Algorithms, (2019), 1–26. doi: 10.1007/s11075-019-00672-3.

[20]

M. Li, C. M. Huang and P. D. Wang, Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algorithms, 74 (2017), 499-525.  doi: 10.1007/s11075-016-0160-5.

[21]

S. Longhi, Fractional Schrödinger equation in optics, Opt. Lett., 40 (2015), 1117. doi: 10.1364/OL.40.001117.

[22]

T. Matsuo and D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), 425-447.  doi: 10.1006/jcph.2001.6775.

[23]

Y. Miyatake and T. Matsuo, Conservative finite difference schemes for the Degasperis-Procesi equation, J. Comput. Appl. Math., 236 (2012), 3728-3740.  doi: 10.1016/j.cam.2011.09.004.

[24]

Y. Miyatake, T. Matsuo and D. Furihata, Invariants-preserving integration of the modified Camassa-Holm equation, Jpn. J. Ind. Appl. Math., 28 (2011), 351-381.  doi: 10.1007/s13160-011-0043-z.

[25]

G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 045206, 7 pp. doi: 10.1088/1751-8113/41/4/045206.

[26]

T. Sogabe and S.-L. Zhang, A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math., 199 (2007), 297-303.  doi: 10.1016/j.cam.2005.07.032.

[27]

H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), 631-644.  doi: 10.1137/0913035.

[28]

H. A. van der Vorst and J. B. Melissen, A Petrov-Galerkin type method for solving ${A}x = b$, where ${A}$ is symmetric complex, IEEE Trans. Mag., 26 (1990), 706-708. 

[29]

D. L. Wang, A. G. Xiao and W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys., 272 (2014), 644-655.  doi: 10.1016/j.jcp.2014.04.047.

[30]

P. D. Wang and C. M. Huang, A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation, Numer. Algorithms, 69 (2015), 625-641.  doi: 10.1007/s11075-014-9917-x.

[31]

P. D. Wang and C. M. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293 (2015), 238-251.  doi: 10.1016/j.jcp.2014.03.037.

[32]

P. D. Wang and C. M. Huang, Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions, Comput. Math. Appl., 71 (2016), 1114-1128.  doi: 10.1016/j.camwa.2016.01.022.

[33]

P. D. Wang and C. M. Huang, Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math., 129 (2018), 137-158.  doi: 10.1016/j.apnum.2018.03.008.

[34]

P. D. Wang, C. M. Huang and L. B. Zhao, Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, J. Comput. Appl. Math., 306 (2016), 231-247.  doi: 10.1016/j.cam.2016.04.017.

[35]

Y. Q. Zhang, X. Liu, M. R. Belić, W. P. Zhong, Y. P. Zhang and M. Xiao, Propagation dynamics of a light beam in a fractional Schrödinger equation, Phys. Rev. Lett., 115 (2015), 180403. doi: 10.1103/PhysRevLett.115.180403.

Figure 1.  Numerical solutions for the case $ \alpha = 2 $ obtained by (LEFT) the linearly implicit scheme (9) and (RIGHT) the nonlinear scheme (14)
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Figure 2.  Numerical solutions for the case $ \alpha = 1.6 $ obtained by (LEFT) the linearly implicit scheme (9) and (RIGHT) the nonlinear scheme (14)
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Figure 3.  Numerical solutions for the case $ \alpha = 1.2 $ obtained by (LEFT) the linearly implicit scheme (9) and (RIGHT) the nonlinear scheme (14)
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Figure 4.  Numerical solutions for the case $ \alpha = 1.6 $ obtained by the linearly implicit scheme (9) with $ N = 61 $
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Figure 5.  Error behaviour obtained by the linearly implicit scheme (9) for the case $ \alpha = 1.6 $: (LEFT) global error, and (RIGHT) error at $ t = 20 $. Errors are measured by $ \max_k | U_k^{(n)} - U_{\text{ref},k}^{(n)}| $, where the reference solution was generated by the nonlinear scheme (14) with $ N = 303 $ and $ \Delta t = 0.001 $
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Figure 6.  Errors of the discrete mass $ \mathcal{M}_ \mathrm{d} ( {\mathit{\boldsymbol{U}}}^{(n)} $ and energy $ \mathcal{H}_ \mathrm{d} ( {\mathit{\boldsymbol{U}}}^{(n+1)}, {\mathit{\boldsymbol{U}}}^{(n)}) $ obtained by the linearly implicit scheme (9) for the case $ \alpha = 2.0 $
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Figure 7.  Errors of the discrete mass $ \mathcal{M}_ \mathrm{d}( {\boldsymbol{U}}^{(n)} $ and energy $ \mathcal{H}_ \mathrm{d} ( {\boldsymbol{U}}^{(n+1)}, {\boldsymbol{U}}^{(n)}) $ obtained by the linearly implicit scheme (9) for the case $ \alpha = 1.6 $
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Figure 8.  Errors of the discrete mass $ \mathcal{M}_ \mathrm{d} ( {\boldsymbol{U}}^{(n)}) $ and energy $ \mathcal{H}_ \mathrm{d} ( {\boldsymbol{U}}^{(n+1)}, {\boldsymbol{U}}^{(n)}) $ obtained by the linearly implicit scheme (9) for the case $ \alpha = 1.2 $
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Figure 9.  Errors of the discrete energy $ \tilde{ \mathcal{H}}_ \mathrm{d} ( {\mathit{\boldsymbol{U}}}^{(n)}) $ obtained by the linearly implicit scheme (9) for the case $ \alpha = 1.6 $
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Figure 10.  The number of iterations for the COCG method to the convergence at each time step. (LEFT) $ N = 101 $ and $ \Delta t = 0.02 $ are fixed, (RIGHT) $ \alpha = 2 $ and $ \Delta t = 0.02 $ are fixed, (BOTTOM) $ \alpha = 2 $ and $ N = 401 $ are fixed
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Figure 11.  The number of iterations for the Bi-CGSTAB method applied to (10). The parameters are set to $ \alpha = 2 $, $ N = 401 $ and $ \Delta t = 0.02 $
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Figure 12.  The number of iterations for the preconditioned COCG method applied to (19) with the matrix $ M = I + \mathrm{i} \Delta t D_\alpha $. The initial condition is set to $ u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10)) $. The parameters are set to $ \alpha = 2 $ and $ N = 401 $. (LEFT) $ \Delta t = 0.01,0.02 $, (RIGHT) $ \Delta t = 0.05 $
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Figure 13.  The number of iterations for the preconditioned COCG method applied to (19) with the matrix $ M = I + \mathrm{i} \Delta t D_\alpha $. The initial condition is set to $ u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(4(x-10)) $. The parameters are set to $ \alpha = 2 $ and $ N = 401 $
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Table 1.  The maximum, minimum and average number of iterations of the preconditioned Bi-CGSTAB method: the time step size is set to $ \Delta t = 0.02 $, and the initial value $ u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10)) $
$ N $ $ 401 $ $ 1001 $ $ 4001 $
maximum $ 3 $ $ 3 $ $ 3 $
minimum $ 3 $ $ 3 $ $ 3 $
average $ 3 $ $ 3 $ $ 3 $
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$ N $ $ 401 $ $ 1001 $ $ 4001 $
maximum $ 3 $ $ 3 $ $ 3 $
minimum $ 3 $ $ 3 $ $ 3 $
average $ 3 $ $ 3 $ $ 3 $
Table 2.  Average CPU time of $ 10 $ simulations at $ T = 8 $ (the cost for obtaining $ {\mathit{\boldsymbol{U}}}^{(1)} $ by the nonlinear scheme (14) is excluded): the time step size is set to $ \Delta t = 0.02 $, and the initial value $ u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10)) $
$ N $ $ 1001 $ $ 2001 $ $ 4001 $ $ 8001 $
CPU time $ 0.794 $ $ 1.524 $ $ 5.615 $ $ 8.223 $
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$ N $ $ 1001 $ $ 2001 $ $ 4001 $ $ 8001 $
CPU time $ 0.794 $ $ 1.524 $ $ 5.615 $ $ 8.223 $
Table 3.  The maximum, minimum and average number of iterations of the preconditioned Bi-CGSTAB method: the time step size is set to $ \Delta t = 0.2 $, and the initial value $ u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(\sqrt{2}(x-10)) $
$ N $ $ 401 $ $ 1001 $ $ 4001 $
maximum $ 6 $ $ 6 $ $ 6 $
minimum $ 5 $ $ 5 $ $ 5 $
average $ 5.020 $ $ 5.020 $ $ 5.020 $
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$ N $ $ 401 $ $ 1001 $ $ 4001 $
maximum $ 6 $ $ 6 $ $ 6 $
minimum $ 5 $ $ 5 $ $ 5 $
average $ 5.020 $ $ 5.020 $ $ 5.020 $
Table 4.  The maximum, minimum and average number of iterations of the preconditioned Bi-CGSTAB method: the time step size is set to $ \Delta t = 0.02 $, and the initial value $ u_0 (x) = 2\exp (0.5 \mathrm{i} x){\rm{sech}}(x-10) $
$ N $ $ 401 $ $ 1001 $ $ 4001 $
maximum $ 4 $ $ 4 $ $ 4 $
minimum $ 3 $ $ 3 $ $ 3 $
average $ 3.286 $ $ 3.291 $ $ 3.323 $
Table Options

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$ N $ $ 401 $ $ 1001 $ $ 4001 $
maximum $ 4 $ $ 4 $ $ 4 $
minimum $ 3 $ $ 3 $ $ 3 $
average $ 3.286 $ $ 3.291 $ $ 3.323 $
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