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A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction
Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
2. | School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan 611731, China |
In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.
References:
[1] |
G. Baruch, G. Fibich and S. Tsynkov,
High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry, J. Comput. Appl. Math., 204 (2007), 477-492.
doi: 10.1016/j.cam.2006.01.048. |
[2] |
G. Baruch, G. Fibich and S. Tsynkov,
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension, J. Comput. Phys., 227 (2007), 820-850.
doi: 10.1016/j.jcp.2007.08.022. |
[3] |
G. Baruch, G. Fibich and S. Tsynkov,
A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media, J. Comput. Phys., 228 (2009), 3789-3815.
doi: 10.1016/j.jcp.2009.02.014. |
[4] |
V. A. Bokil, Y. Cheng, Y. Jiang and F. Li,
Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media, J. Comput. Phys., 350 (2017), 420-452.
doi: 10.1016/j.jcp.2017.08.009. |
[5] |
R. W. Boyd, Nonlinear Optics, Elsevier/Academic Press, Amsterdam, 2008.
![]() |
[6] |
E. Centeno and D. Felbacq,
Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity, Phys. Rev. B, 62 (2000), 7683-7686.
doi: 10.1103/PhysRevB.62.R7683. |
[7] |
W. Chen and D. L. Mills,
Optical response of a nonlinear dielectric film, Phys. Rev. B, 35 (1987), 524-532.
doi: 10.1103/PhysRevB.35.524. |
[8] |
W. Chen and D. L. Mills,
Optical response of nonlinear multilayer structures: Bilayers and superlattices, Phys. Rev. B, 36 (1987), 524-532.
doi: 10.1103/PhysRevB.36.6269. |
[9] |
W. Dai and R. Nassar,
Compact ADI method for solving parabolic differential equations, Numer. Methods Partial Differential Equations, 18 (2002), 129-142.
doi: 10.1002/num.1037. |
[10] |
W. Dai and R. Nassar,
A new ADI scheme for solving three-dimensional parabolic equations with first-order derivatives and variable coefficients, J. Comput. Anal. Appl., 2 (2000), 293-308.
doi: 10.1023/A:1010108620966. |
[11] |
G. Evequoz and T. Weth,
Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728.
doi: 10.1016/j.aim.2015.04.017. |
[12] |
G. Evéquoz,
Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane, Analysis (Berlin), 37 (2017), 55-68.
doi: 10.1515/anly-2016-0023. |
[13] |
G. Fibich, The Nonlinear Schrödinger Equation. Singular Solutions and Optical Collapse, Applied Mathematical Sciences, 192, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12748-4. |
[14] |
G. Fibich and S. Tsynkov,
High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001), 632-677.
doi: 10.1006/jcph.2001.6800. |
[15] |
G. Fibich and S. Tsynkov,
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005), 183-224.
doi: 10.1016/j.jcp.2005.04.015. |
[16] |
R. Guo, K. Wang and L. Xu,
Efficient finite difference methods for acoustic scattering from circular cylindrical obstacle, Int. J. Numer. Anal. Model., 13 (2016), 986-1002.
|
[17] |
X. He and K. Wang,
Uniformly convergent novel finite difference methods for singularly perturbed reaction-diffusion equations, Numer. Methods Partial Differential Equations, 35 (2019), 2120-2148.
doi: 10.1002/num.22405. |
[18] |
T. A. Laine and A. T. Friberg,
Self-guided waves and exact solutions of the nonlinear Helmholtz equation, J. Opt. Soc. Amer. B Opt. Phys., 17 (2000), 751-757.
doi: 10.1364/JOSAB.17.000751. |
[19] |
R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz equations, Z. Angew. Math. Phys., 68 (2017), 19pp.
doi: 10.1007/s00033-017-0859-8. |
[20] |
G. I. Stegeman and M. Segev,
Optical spatial solitons and their interactions: Universality and diversity, Science, 286 (1999), 1518-1523.
doi: 10.1126/science.286.5444.1518. |
[21] |
J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004.
doi: 10.1137/1.9780898717938. |
[22] |
A. Suryanto, E. van Groesen and M. Hammer,
Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with a Defect, J. Nonlinear Optical Phys. Materials, 12 (2003), 187-204.
doi: 10.1142/S0218863503001328. |
[23] |
A. Suryanto, E. van Groesen and M. Hammer, A finite element scheme to study the nonlinear optical response of a finite grating without and with defect, Optical and Quantum Electronics, 35 (2003), 313-332.
doi: 10.1023/A:1022901201632. |
[24] |
K. Wang and Y. S. Wong,
Error correction method for Navier-Stokes equations at high Reynolds numbers, J. Comput. Phys., 255 (2013), 245-265.
doi: 10.1016/j.jcp.2013.07.042. |
[25] |
K. Wang and Y. S. Wong,
Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model., 11 (2014), 787-815.
|
[26] |
K. Wang and Y. S. Wong,
Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?, Commun. Comput. Phys., 21 (2017), 490-514.
doi: 10.4208/cicp.OA-2016-0057. |
[27] |
K. Wang, Y. S. Wong and J. Deng,
Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys., 17 (2015), 779-807.
doi: 10.4208/cicp.110214.101014a. |
[28] |
K. Wang, Y. S. Wong and J. Huang,
Analysis of pollution-free approaches for multi-dimensional Helmholtz equations, Int. J. Numer. Anal. Model., 16 (2019), 412-435.
|
[29] |
H. Wu and J. Zou,
Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 56 (2018), 1338-1359.
doi: 10.1137/17M111314X. |
[30] |
Z. Xu and G. Bao,
A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects, Journal of the Optical Society of America(A), 27 (2010), 2347-2353.
doi: 10.1364/JOSAA.27.002347. |
[31] |
L. Yuan and Y. Y. Lu,
Robust iterative method for nonlinear Helmholtz equation, J. Comput. Phys., 343 (2017), 1-9.
doi: 10.1016/j.jcp.2017.04.046. |
[32] |
S. Zhai, X. Feng and Y. He,
A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation, J. Sci. Comput., 54 (2013), 97-120.
doi: 10.1007/s10915-012-9607-6. |
[33] |
S. Zhai, X. Feng and Y. He,
A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation, Appl. Math. Comput., 230 (2014), 9-26.
doi: 10.1016/j.amc.2013.12.096. |
show all references
References:
[1] |
G. Baruch, G. Fibich and S. Tsynkov,
High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry, J. Comput. Appl. Math., 204 (2007), 477-492.
doi: 10.1016/j.cam.2006.01.048. |
[2] |
G. Baruch, G. Fibich and S. Tsynkov,
High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension, J. Comput. Phys., 227 (2007), 820-850.
doi: 10.1016/j.jcp.2007.08.022. |
[3] |
G. Baruch, G. Fibich and S. Tsynkov,
A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media, J. Comput. Phys., 228 (2009), 3789-3815.
doi: 10.1016/j.jcp.2009.02.014. |
[4] |
V. A. Bokil, Y. Cheng, Y. Jiang and F. Li,
Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media, J. Comput. Phys., 350 (2017), 420-452.
doi: 10.1016/j.jcp.2017.08.009. |
[5] |
R. W. Boyd, Nonlinear Optics, Elsevier/Academic Press, Amsterdam, 2008.
![]() |
[6] |
E. Centeno and D. Felbacq,
Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity, Phys. Rev. B, 62 (2000), 7683-7686.
doi: 10.1103/PhysRevB.62.R7683. |
[7] |
W. Chen and D. L. Mills,
Optical response of a nonlinear dielectric film, Phys. Rev. B, 35 (1987), 524-532.
doi: 10.1103/PhysRevB.35.524. |
[8] |
W. Chen and D. L. Mills,
Optical response of nonlinear multilayer structures: Bilayers and superlattices, Phys. Rev. B, 36 (1987), 524-532.
doi: 10.1103/PhysRevB.36.6269. |
[9] |
W. Dai and R. Nassar,
Compact ADI method for solving parabolic differential equations, Numer. Methods Partial Differential Equations, 18 (2002), 129-142.
doi: 10.1002/num.1037. |
[10] |
W. Dai and R. Nassar,
A new ADI scheme for solving three-dimensional parabolic equations with first-order derivatives and variable coefficients, J. Comput. Anal. Appl., 2 (2000), 293-308.
doi: 10.1023/A:1010108620966. |
[11] |
G. Evequoz and T. Weth,
Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728.
doi: 10.1016/j.aim.2015.04.017. |
[12] |
G. Evéquoz,
Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane, Analysis (Berlin), 37 (2017), 55-68.
doi: 10.1515/anly-2016-0023. |
[13] |
G. Fibich, The Nonlinear Schrödinger Equation. Singular Solutions and Optical Collapse, Applied Mathematical Sciences, 192, Springer, Cham, 2015.
doi: 10.1007/978-3-319-12748-4. |
[14] |
G. Fibich and S. Tsynkov,
High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001), 632-677.
doi: 10.1006/jcph.2001.6800. |
[15] |
G. Fibich and S. Tsynkov,
Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005), 183-224.
doi: 10.1016/j.jcp.2005.04.015. |
[16] |
R. Guo, K. Wang and L. Xu,
Efficient finite difference methods for acoustic scattering from circular cylindrical obstacle, Int. J. Numer. Anal. Model., 13 (2016), 986-1002.
|
[17] |
X. He and K. Wang,
Uniformly convergent novel finite difference methods for singularly perturbed reaction-diffusion equations, Numer. Methods Partial Differential Equations, 35 (2019), 2120-2148.
doi: 10.1002/num.22405. |
[18] |
T. A. Laine and A. T. Friberg,
Self-guided waves and exact solutions of the nonlinear Helmholtz equation, J. Opt. Soc. Amer. B Opt. Phys., 17 (2000), 751-757.
doi: 10.1364/JOSAB.17.000751. |
[19] |
R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz equations, Z. Angew. Math. Phys., 68 (2017), 19pp.
doi: 10.1007/s00033-017-0859-8. |
[20] |
G. I. Stegeman and M. Segev,
Optical spatial solitons and their interactions: Universality and diversity, Science, 286 (1999), 1518-1523.
doi: 10.1126/science.286.5444.1518. |
[21] |
J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004.
doi: 10.1137/1.9780898717938. |
[22] |
A. Suryanto, E. van Groesen and M. Hammer,
Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with a Defect, J. Nonlinear Optical Phys. Materials, 12 (2003), 187-204.
doi: 10.1142/S0218863503001328. |
[23] |
A. Suryanto, E. van Groesen and M. Hammer, A finite element scheme to study the nonlinear optical response of a finite grating without and with defect, Optical and Quantum Electronics, 35 (2003), 313-332.
doi: 10.1023/A:1022901201632. |
[24] |
K. Wang and Y. S. Wong,
Error correction method for Navier-Stokes equations at high Reynolds numbers, J. Comput. Phys., 255 (2013), 245-265.
doi: 10.1016/j.jcp.2013.07.042. |
[25] |
K. Wang and Y. S. Wong,
Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model., 11 (2014), 787-815.
|
[26] |
K. Wang and Y. S. Wong,
Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?, Commun. Comput. Phys., 21 (2017), 490-514.
doi: 10.4208/cicp.OA-2016-0057. |
[27] |
K. Wang, Y. S. Wong and J. Deng,
Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys., 17 (2015), 779-807.
doi: 10.4208/cicp.110214.101014a. |
[28] |
K. Wang, Y. S. Wong and J. Huang,
Analysis of pollution-free approaches for multi-dimensional Helmholtz equations, Int. J. Numer. Anal. Model., 16 (2019), 412-435.
|
[29] |
H. Wu and J. Zou,
Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 56 (2018), 1338-1359.
doi: 10.1137/17M111314X. |
[30] |
Z. Xu and G. Bao,
A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects, Journal of the Optical Society of America(A), 27 (2010), 2347-2353.
doi: 10.1364/JOSAA.27.002347. |
[31] |
L. Yuan and Y. Y. Lu,
Robust iterative method for nonlinear Helmholtz equation, J. Comput. Phys., 343 (2017), 1-9.
doi: 10.1016/j.jcp.2017.04.046. |
[32] |
S. Zhai, X. Feng and Y. He,
A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation, J. Sci. Comput., 54 (2013), 97-120.
doi: 10.1007/s10915-012-9607-6. |
[33] |
S. Zhai, X. Feng and Y. He,
A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation, Appl. Math. Comput., 230 (2014), 9-26.
doi: 10.1016/j.amc.2013.12.096. |








100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
100 | 200 | 400 | 800 | 1600 | |
SFD | 2.14 | 1.05 | 2.69e-1 | 6.71e-2 | 1.67e-3 |
FV[2] | 1.59 | 5.03e-1 | 1.29e-1 | 3.23e-2 | 8.09e-3 |
CFD | 5.38e-1 | 3.70e-2 | 3.27e-3 | 6.67e-4 | 2.72e-4 |
Scheme (29) | 1.26e-3 | 2.99e-4 | 7.43e-5 | 1.89e-5 | 5.16e-6 |
Scheme (30) | 2.16e-4 | 5.68e-5 | 1.43e-5 | 3.68e-6 | 1.16e-6 |
SFD | 2.31 | 2.13 | 1.80 | 5.33e-1 | 1.34e-1 |
FV[2] | 2.00 | 2.00 | 9.76e-1 | 2.60e-1 | 6.52e-2 |
CFD | 2.17 | 1.02 | 7.16e-2 | 5.46e-3 | 8.00e-4 |
Scheme (29) | 8.56e-3 | 1.57e-3 | 3.75e-4 | 9.57e-5 | 2.70e-5 |
Scheme (30) | 1.29e-3 | 3.16e-4 | 7.56e-5 | 1.87e-5 | 4.95e-6 |
SFD | 1.24 | 2.35 | 2.13 | 2.03 | 1.03 |
FV[2] | - | 2.00 | 1.99 | 1.70 | 5.16e-1 |
CFD | 1.22 | 2.36 | 1.76 | 1.40e-1 | 9.86e-3 |
Scheme (29) | 1.12e-2 | 9.70e-3 | 1.80e-3 | 4.49e-4 | 1.31e-4 |
Scheme (30) | 5.64e-3 | 2.68e-3 | 6.86e-4 | 1.05e-4 | 2.61e-5 |
SFD | 1.07 | 1.05 | 2.32 | 2.29 | 2.02 |
FV[2] | - | - | 2.00 | 1.98 | 1.97 |
CFD | 1.04 | 1.21 | 2.31 | 1.99 | 0.29 |
Scheme (29) | 7.38e-3 | 8.68e-3 | 4.92e-3 | 1.99e-3 | 1.52e-3 |
Scheme (30) | 1.47e-3 | 1.05e-3 | 2.48e-4 | 2.56e-4 | 2.04e-4 |
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |
10 | 20 | 40 | 80 | 160 | 320 | 640 | 1280 | |
Frozen-nonlinearity | 5 | 5 | 6 | 7 | 9 | 12 | 17 | 38 |
Error Correction | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 |
Modified Newton | 5 | 6 | 7 | 8 | 10 | 14 | 22 | - |
Newton's method | 4 | 4 | 5 | 5 | 6 | 8 | 11 | - |
Frozen-nonlinearity | 5 | 6 | 7 | 9 | 12 | 19 | 45 | - |
Error Correction | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 9 |
Modified Newton | 6 | 7 | 8 | 10 | 14 | 23 | - | - |
Newton's method | 4 | 5 | 5 | 6 | 8 | 11 | - | - |
Frozen-nonlinearity | 6 | 8 | 9 | 13 | 22 | 55 | - | - |
Error Correction | 4 | 4 | 5 | 5 | 6 | 8 | 13 | - |
Modified Newton | 7 | 9 | 10 | 16 | 25 | - | - | - |
Newton's method | 5 | 5 | 6 | 8 | 10 | - | - | - |
Frozen-nonlinearity | 7 | 9 | 12 | 18 | 35 | - | - | - |
Error Correction | 4 | 5 | 5 | 6 | 7 | 10 | - | - |
Modified Newton | 8 | 9 | 14 | 20 | 39 | - | - | - |
Newton's method | 5 | 6 | 7 | 10 | - | - | - | - |
Frozen-nonlinearity | 8 | 10 | 14 | 20 | 89 | - | - | - |
Error Correction | 5 | 5 | 6 | 7 | 9 | - | - | - |
Modified Newton | 9 | 11 | 17 | 25 | - | - | - | - |
Newton's method | 5 | 6 | 8 | 11 | - | - | - | - |
Frozen-nonlinearity | 9 | 10 | 16 | 35 | - | - | - | - |
Error Correction | 5 | 6 | 6 | 8 | 12 | - | - | - |
Modified Newton | 10 | 13 | 18 | 36 | - | - | - | - |
Newton's method | 6 | 7 | 9 | - | - | - | - | - |
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