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An adaptive finite element DtN method for the three-dimensional acoustic scattering problem
A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
1. | School of Mathematical Science, Beijing Normal University, Beijing l00875, China |
2. | Beijing Computational Science Research Center, Beijing 100193, China, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA |
In this paper, we present and study $ C^1 $ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $ k $ ($ \ge 3 $) for one-dimen\-sional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $ 2k-2 $ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree $ k+1 $ in each element, the first-order derivative approximation is superconvergent at all interior $ k-2 $ Lobatto points, and the second-order derivative approximation is superconvergent at $ k-1 $ Gauss points, with an order of $ k+2 $, $ k+1 $, and $ k $, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in $ H^2 $, $ H^1 $, and $ L^2 $ norms. All theoretical findings are confirmed by numerical experiments.
References:
[1] |
S. Adjerid and T. C. Massey,
Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3331-3346.
doi: 10.1016/j.cma.2005.06.017. |
[2] |
S. Adjerid and T. Weinhart,
Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3113-3129.
doi: 10.1016/j.cma.2009.05.016. |
[3] |
S. Adjerid and T. Weinhart,
Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems,, Math. Comp., 80 (2011), 1335-1367.
doi: 10.1090/S0025-5718-2011-02460-9. |
[4] |
I. Babu$ \rm\check{s} $ka, T. Strouboulis, C. S. Upadhyay and S. K. Gangaraj,
Computer-based proof of the existence of superconvergence points in the finite element method: Superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations, Numer. Meth. PDEs, 12 (1996), 347-392.
doi: 10.1002/num.1690120303. |
[5] |
S. K. Bhal and P. Danumjaya,
A Fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity, J. Anal., 27 (2019), 377-390.
doi: 10.1007/s41478-018-0082-9. |
[6] |
B. Bialecki,
Superconvergence of the orthogonal spline collocation solution of Poisson's equation,, Numerical Methods for Partial Differential Equations, 15 (1999), 285-303.
doi: 10.1002/(SICI)1098-2426(199905)15:3<285::AID-NUM2>3.0.CO;2-1. |
[7] |
J. H. Bramble and A. H. Schatz,
High order local accuracy by averaging in the finite element method, Math. Comp., 31 (1997), 94-111.
doi: 10.1090/S0025-5718-1977-0431744-9. |
[8] |
Z. Q. Cai,
On the finite volume element method, Numer. Math., 58 (1991), 713-735.
doi: 10.1007/BF01385651. |
[9] |
W. Cao, C.-W. Shu, Y. Yang and Z. Zhang,
Superconvergence of Discontinuous Galerkin method for nonlinear hyperbolic equations, SIAM. J. Numer. Anal., 56 (2018), 732-765.
doi: 10.1137/17M1128605. |
[10] |
W. Cao and Z. Zhang,
Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations, Math. Comp., 85 (2016), 63-84.
doi: 10.1090/mcom/2975. |
[11] |
W. Cao, Z. Zhang and Q. Zou,
Superconvergence of any order finite volume schemes for 1D general elliptic equations, J. Sci. Comput., 56 (2013), 566-590.
doi: 10.1007/s10915-013-9691-2. |
[12] |
W. Cao, Z. Zhang and Q. Zou,
Superconvergence of Discontinuous Galerkin method for linear hyperbolic equations, SIAM. J. Numer. Anal., 52 (2014), 2555-2573.
doi: 10.1137/130946873. |
[13] |
W. Cao, Z. Zhang and Q. Zou,
Is $2k$-conjecture valid for finite volume methods?, SIAM. J. Numer. Anal., 53 (2015), 942-962.
doi: 10.1137/130936178. |
[14] |
C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Hunan, China, 2001.
![]() |
[15] |
C. Chen and S. Hu,
The highest order superconvergence for bi-$k$ degree rectangular elements at nodes- a proof of $2k$-conjecture,, Math. Comp., 82 (2013), 1337-1355.
doi: 10.1090/S0025-5718-2012-02653-6. |
[16] |
C. Chen and Y. Huang, High Accuracy Theory of Finite Elements, Hunan Science and Technology Press, Hunan, China, 1995.
![]() |
[17] |
Y. Cheng and C.-W. Shu,
Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47 (2010), 4044-4072.
doi: 10.1137/090747701. |
[18] |
S.-H. Chou and X. Ye,
Superconvergence of finite volume methods for the second order elliptic problem, Comput. Methods Appl. Mech. Eng., 196 (2007), 3706-3712.
doi: 10.1016/j.cma.2006.10.025. |
[19] |
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second edition, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984.
![]() ![]() |
[20] |
R. E. Ewing, R. D. Lazarov and J. Wang,
Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28 (1991), 1015-1029.
doi: 10.1137/0728054. |
[21] |
M. K$ \rm\check{r} $í$ \rm\check{z} $ek and P. Neittaanm$\ddot{a}$ki,
On superconvergence techniques, Acta Appl. Math., 9 (1987), 175-198.
doi: 10.1007/BF00047538. |
[22] |
M. K$ \rm\check{r} $í$ \rm\check{z} $ek, P. Neittaanm$\ddot{a}$ki and R. Stenberg (Eds.), Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series Vol. 196, Marcel Dekker, Inc., New York, 1997. |
[23] |
Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, P.R. China, 1996.
![]() |
[24] |
A. H. Schatz, I. H. Sloan and L. B. Wahlbin,
Superconvergence in finite element methods and meshes which are symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), 505-521.
doi: 10.1137/0733027. |
[25] |
J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-540-71041-7. |
[26] |
V. Thomée,
High order local approximation to derivatives in the finite element method, Math. Comp., 31 (1997), 652-660.
doi: 10.1090/S0025-5718-1977-0438664-4. |
[27] |
L. B. Wahlbin, Superconvergence In Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605. Spring, Berlin, 1995.
doi: 10.1007/BFb0096835. |
[28] |
Z. Xie and Z. Zhang,
Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D, Math. Comp., 79 (2010), 35-45.
doi: 10.1090/S0025-5718-09-02297-2. |
[29] |
J. Xu and Q. Zou,
Analysis of linear and quadratic simplitical finite volume methods for elliptic equations, Numer. Math., 111 (2009), 469-492.
doi: 10.1007/s00211-008-0189-z. |
[30] |
Y. Yang and C.-W. Shu,
Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal., 50 (2012), 3110-3133.
doi: 10.1137/110857647. |
[31] |
Z. Zhang,
Superconvergence points of polynomial spectral interpolation,, SIAM J. Numer. Anal., 50 (2012), 2966-2985.
doi: 10.1137/120861291. |
[32] |
Z. Zhang,
Superconvergence of a Chebyshev spectral collocation method, J. Sci. Comput., 34 (2008), 237-246.
doi: 10.1007/s10915-007-9163-7. |
[33] |
Q. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Method, Hunan Science and Technology Press, Hunan, China, 1989.
![]() |
show all references
References:
[1] |
S. Adjerid and T. C. Massey,
Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3331-3346.
doi: 10.1016/j.cma.2005.06.017. |
[2] |
S. Adjerid and T. Weinhart,
Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3113-3129.
doi: 10.1016/j.cma.2009.05.016. |
[3] |
S. Adjerid and T. Weinhart,
Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems,, Math. Comp., 80 (2011), 1335-1367.
doi: 10.1090/S0025-5718-2011-02460-9. |
[4] |
I. Babu$ \rm\check{s} $ka, T. Strouboulis, C. S. Upadhyay and S. K. Gangaraj,
Computer-based proof of the existence of superconvergence points in the finite element method: Superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's, and the elasticity equations, Numer. Meth. PDEs, 12 (1996), 347-392.
doi: 10.1002/num.1690120303. |
[5] |
S. K. Bhal and P. Danumjaya,
A Fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity, J. Anal., 27 (2019), 377-390.
doi: 10.1007/s41478-018-0082-9. |
[6] |
B. Bialecki,
Superconvergence of the orthogonal spline collocation solution of Poisson's equation,, Numerical Methods for Partial Differential Equations, 15 (1999), 285-303.
doi: 10.1002/(SICI)1098-2426(199905)15:3<285::AID-NUM2>3.0.CO;2-1. |
[7] |
J. H. Bramble and A. H. Schatz,
High order local accuracy by averaging in the finite element method, Math. Comp., 31 (1997), 94-111.
doi: 10.1090/S0025-5718-1977-0431744-9. |
[8] |
Z. Q. Cai,
On the finite volume element method, Numer. Math., 58 (1991), 713-735.
doi: 10.1007/BF01385651. |
[9] |
W. Cao, C.-W. Shu, Y. Yang and Z. Zhang,
Superconvergence of Discontinuous Galerkin method for nonlinear hyperbolic equations, SIAM. J. Numer. Anal., 56 (2018), 732-765.
doi: 10.1137/17M1128605. |
[10] |
W. Cao and Z. Zhang,
Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations, Math. Comp., 85 (2016), 63-84.
doi: 10.1090/mcom/2975. |
[11] |
W. Cao, Z. Zhang and Q. Zou,
Superconvergence of any order finite volume schemes for 1D general elliptic equations, J. Sci. Comput., 56 (2013), 566-590.
doi: 10.1007/s10915-013-9691-2. |
[12] |
W. Cao, Z. Zhang and Q. Zou,
Superconvergence of Discontinuous Galerkin method for linear hyperbolic equations, SIAM. J. Numer. Anal., 52 (2014), 2555-2573.
doi: 10.1137/130946873. |
[13] |
W. Cao, Z. Zhang and Q. Zou,
Is $2k$-conjecture valid for finite volume methods?, SIAM. J. Numer. Anal., 53 (2015), 942-962.
doi: 10.1137/130936178. |
[14] |
C. Chen, Structure Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Hunan, China, 2001.
![]() |
[15] |
C. Chen and S. Hu,
The highest order superconvergence for bi-$k$ degree rectangular elements at nodes- a proof of $2k$-conjecture,, Math. Comp., 82 (2013), 1337-1355.
doi: 10.1090/S0025-5718-2012-02653-6. |
[16] |
C. Chen and Y. Huang, High Accuracy Theory of Finite Elements, Hunan Science and Technology Press, Hunan, China, 1995.
![]() |
[17] |
Y. Cheng and C.-W. Shu,
Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal., 47 (2010), 4044-4072.
doi: 10.1137/090747701. |
[18] |
S.-H. Chou and X. Ye,
Superconvergence of finite volume methods for the second order elliptic problem, Comput. Methods Appl. Mech. Eng., 196 (2007), 3706-3712.
doi: 10.1016/j.cma.2006.10.025. |
[19] |
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Second edition, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984.
![]() ![]() |
[20] |
R. E. Ewing, R. D. Lazarov and J. Wang,
Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28 (1991), 1015-1029.
doi: 10.1137/0728054. |
[21] |
M. K$ \rm\check{r} $í$ \rm\check{z} $ek and P. Neittaanm$\ddot{a}$ki,
On superconvergence techniques, Acta Appl. Math., 9 (1987), 175-198.
doi: 10.1007/BF00047538. |
[22] |
M. K$ \rm\check{r} $í$ \rm\check{z} $ek, P. Neittaanm$\ddot{a}$ki and R. Stenberg (Eds.), Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series Vol. 196, Marcel Dekker, Inc., New York, 1997. |
[23] |
Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, P.R. China, 1996.
![]() |
[24] |
A. H. Schatz, I. H. Sloan and L. B. Wahlbin,
Superconvergence in finite element methods and meshes which are symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), 505-521.
doi: 10.1137/0733027. |
[25] |
J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-540-71041-7. |
[26] |
V. Thomée,
High order local approximation to derivatives in the finite element method, Math. Comp., 31 (1997), 652-660.
doi: 10.1090/S0025-5718-1977-0438664-4. |
[27] |
L. B. Wahlbin, Superconvergence In Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605. Spring, Berlin, 1995.
doi: 10.1007/BFb0096835. |
[28] |
Z. Xie and Z. Zhang,
Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D, Math. Comp., 79 (2010), 35-45.
doi: 10.1090/S0025-5718-09-02297-2. |
[29] |
J. Xu and Q. Zou,
Analysis of linear and quadratic simplitical finite volume methods for elliptic equations, Numer. Math., 111 (2009), 469-492.
doi: 10.1007/s00211-008-0189-z. |
[30] |
Y. Yang and C.-W. Shu,
Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal., 50 (2012), 3110-3133.
doi: 10.1137/110857647. |
[31] |
Z. Zhang,
Superconvergence points of polynomial spectral interpolation,, SIAM J. Numer. Anal., 50 (2012), 2966-2985.
doi: 10.1137/120861291. |
[32] |
Z. Zhang,
Superconvergence of a Chebyshev spectral collocation method, J. Sci. Comput., 34 (2008), 237-246.
doi: 10.1007/s10915-007-9163-7. |
[33] |
Q. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Method, Hunan Science and Technology Press, Hunan, China, 1989.
![]() |
error | order | error | order | error | order | error | order | error | order | ||
2 | 8.03e-04 | - | 6.11e-03 | - | - | - | 7.59e-03 | - | 1.31e-01 | - | |
4 | 7.02e-05 | 3.49 | 4.92e-04 | 3.61 | - | - | 5.61e-04 | 3.73 | 1.66e-02 | 2.98 | |
3 | 8 | 4.59e-06 | 3.95 | 3.02e-05 | 4.04 | - | - | 3.73e-05 | 3.92 | 2.18e-03 | 2.93 |
16 | 2.91e-07 | 4.04 | 1.90e-06 | 4.05 | - | - | 2.66e-06 | 3.87 | 2.67e-04 | 3.03 | |
32 | 1.80e-08 | 4.00 | 1.18e-07 | 3.99 | - | - | 1.77e-07 | 3.90 | 3.38e-05 | 2.98 | |
2 | 2.88e-05 | - | 2.23e-05 | - | 7.54e-05 | - | 8.72e-04 | - | 1.30e-02 | - | |
4 | 4.25e-07 | 6.10 | 2.47e-07 | 6.51 | 1.36e-06 | 5.81 | 2.44e-05 | 5.17 | 8.88e-04 | 3.88 | |
4 | 8 | 6.53e-09 | 6.21 | 5.38e-09 | 5.69 | 2.14e-08 | 6.17 | 7.91e-07 | 5.10 | 5.47e-05 | 4.02 |
16 | 1.04e-10 | 5.96 | 1.04e-10 | 5.67 | 3.45e-10 | 5.94 | 2.64e-08 | 4.89 | 3.73e-06 | 3.87 | |
32 | 1.62e-12 | 6.00 | 1.84e-12 | 5.83 | 5.50e-12 | 5.97 | 8.62e-10 | 4.94 | 2.30e-07 | 4.02 |
error | order | error | order | error | order | error | order | error | order | ||
2 | 8.03e-04 | - | 6.11e-03 | - | - | - | 7.59e-03 | - | 1.31e-01 | - | |
4 | 7.02e-05 | 3.49 | 4.92e-04 | 3.61 | - | - | 5.61e-04 | 3.73 | 1.66e-02 | 2.98 | |
3 | 8 | 4.59e-06 | 3.95 | 3.02e-05 | 4.04 | - | - | 3.73e-05 | 3.92 | 2.18e-03 | 2.93 |
16 | 2.91e-07 | 4.04 | 1.90e-06 | 4.05 | - | - | 2.66e-06 | 3.87 | 2.67e-04 | 3.03 | |
32 | 1.80e-08 | 4.00 | 1.18e-07 | 3.99 | - | - | 1.77e-07 | 3.90 | 3.38e-05 | 2.98 | |
2 | 2.88e-05 | - | 2.23e-05 | - | 7.54e-05 | - | 8.72e-04 | - | 1.30e-02 | - | |
4 | 4.25e-07 | 6.10 | 2.47e-07 | 6.51 | 1.36e-06 | 5.81 | 2.44e-05 | 5.17 | 8.88e-04 | 3.88 | |
4 | 8 | 6.53e-09 | 6.21 | 5.38e-09 | 5.69 | 2.14e-08 | 6.17 | 7.91e-07 | 5.10 | 5.47e-05 | 4.02 |
16 | 1.04e-10 | 5.96 | 1.04e-10 | 5.67 | 3.45e-10 | 5.94 | 2.64e-08 | 4.89 | 3.73e-06 | 3.87 | |
32 | 1.62e-12 | 6.00 | 1.84e-12 | 5.83 | 5.50e-12 | 5.97 | 8.62e-10 | 4.94 | 2.30e-07 | 4.02 |
error | order | error | order | error | order | error | order | error | order | ||
2 | 5.25e-03 | - | 1.36e-02 | - | - | - | 1.44e-02 | - | 8.32e-02 | - | |
4 | 2.88e-04 | 4.13 | 7.26e-04 | 4.18 | - | - | 8.35e-04 | 4.06 | 1.16e-02 | 2.84 | |
3 | 8 | 1.82e-05 | 4.07 | 4.66e-05 | 4.05 | - | - | 5.89e-05 | 3.91 | 1.61e-03 | 2.85 |
16 | 1.16e-06 | 3.94 | 2.91e-06 | 3.96 | - | - | 4.01e-06 | 3.84 | 1.91e-04 | 3.08 | |
32 | 7.18e-08 | 4.01 | 1.81e-07 | 4.01 | - | - | 2.65e-07 | 3.92 | 2.45e-05 | 2.96 | |
2 | 1.32e-05 | - | 1.04e-04 | - | 1.98e-04 | - | 9.54e-04 | - | 7.12e-03 | - | |
4 | 2.92e-07 | 5.48 | 1.79e-06 | 5.85 | 3.14e-06 | 5.96 | 3.32e-05 | 4.83 | 5.77e-04 | 3.63 | |
4 | 8 | 4.62e-09 | 5.97 | 2.80e-08 | 5.99 | 5.09e-08 | 5.94 | 1.05e-06 | 4.98 | 3.89e-05 | 3.89 |
16 | 7.56e-11 | 6.06 | 4.40e-10 | 6.12 | 8.61e-10 | 6.01 | 3.40e-08 | 5.04 | 2.46e-06 | 3.98 | |
32 | 1.18e-12 | 6.08 | 6.87e-12 | 6.08 | 1.28e-11 | 6.15 | 1.05e-09 | 5.08 | 1.52e-07 | 4.02 |
error | order | error | order | error | order | error | order | error | order | ||
2 | 5.25e-03 | - | 1.36e-02 | - | - | - | 1.44e-02 | - | 8.32e-02 | - | |
4 | 2.88e-04 | 4.13 | 7.26e-04 | 4.18 | - | - | 8.35e-04 | 4.06 | 1.16e-02 | 2.84 | |
3 | 8 | 1.82e-05 | 4.07 | 4.66e-05 | 4.05 | - | - | 5.89e-05 | 3.91 | 1.61e-03 | 2.85 |
16 | 1.16e-06 | 3.94 | 2.91e-06 | 3.96 | - | - | 4.01e-06 | 3.84 | 1.91e-04 | 3.08 | |
32 | 7.18e-08 | 4.01 | 1.81e-07 | 4.01 | - | - | 2.65e-07 | 3.92 | 2.45e-05 | 2.96 | |
2 | 1.32e-05 | - | 1.04e-04 | - | 1.98e-04 | - | 9.54e-04 | - | 7.12e-03 | - | |
4 | 2.92e-07 | 5.48 | 1.79e-06 | 5.85 | 3.14e-06 | 5.96 | 3.32e-05 | 4.83 | 5.77e-04 | 3.63 | |
4 | 8 | 4.62e-09 | 5.97 | 2.80e-08 | 5.99 | 5.09e-08 | 5.94 | 1.05e-06 | 4.98 | 3.89e-05 | 3.89 |
16 | 7.56e-11 | 6.06 | 4.40e-10 | 6.12 | 8.61e-10 | 6.01 | 3.40e-08 | 5.04 | 2.46e-06 | 3.98 | |
32 | 1.18e-12 | 6.08 | 6.87e-12 | 6.08 | 1.28e-11 | 6.15 | 1.05e-09 | 5.08 | 1.52e-07 | 4.02 |
error | order | error | order | error | order | error | order | ||
2 | 3.93e-02 | - | 5.57e-03 | - | 1.12e-01 | - | 8.32e-02 | - | |
4 | 5.15e-03 | 2.91 | 3.59e-04 | 3.94 | 1.36e-02 | 3.00 | 1.07e-02 | 2.93 | |
3 | 8 | 6.47e-04 | 3.00 | 2.23e-05 | 3.99 | 1.72e-03 | 3.04 | 1.34e-03 | 3.03 |
16 | 8.12e-05 | 3.04 | 1.40e-06 | 4.02 | 2.17e-04 | 2.96 | 1.70e-04 | 2.96 | |
32 | 1.01e-05 | 2.99 | 8.75e-08 | 4.01 | 2.70e-05 | 3.01 | 2.12e-05 | 3.03 | |
2 | 3.47e-03 | - | 2.23e-04 | - | 9.86e-03 | - | 8.27e-03 | - | |
4 | 2.24e-04 | 3.97 | 7.13e-06 | 4.99 | 6.63e-04 | 3.88 | 5.20e-04 | 4.02 | |
4 | 8 | 1.40e-05 | 4.12 | 2.30e-07 | 5.03 | 4.14e-05 | 4.00 | 3.30e-05 | 3.94 |
16 | 8.80e-07 | 3.98 | 7.07e-09 | 5.01 | 2.59e-06 | 4.08 | 2.07e-06 | 4.05 | |
32 | 5.50e-08 | 4.00 | 2.22e-10 | 5.02 | 1.62e-07 | 4.05 | 1.30e-07 | 4.04 |
error | order | error | order | error | order | error | order | ||
2 | 3.93e-02 | - | 5.57e-03 | - | 1.12e-01 | - | 8.32e-02 | - | |
4 | 5.15e-03 | 2.91 | 3.59e-04 | 3.94 | 1.36e-02 | 3.00 | 1.07e-02 | 2.93 | |
3 | 8 | 6.47e-04 | 3.00 | 2.23e-05 | 3.99 | 1.72e-03 | 3.04 | 1.34e-03 | 3.03 |
16 | 8.12e-05 | 3.04 | 1.40e-06 | 4.02 | 2.17e-04 | 2.96 | 1.70e-04 | 2.96 | |
32 | 1.01e-05 | 2.99 | 8.75e-08 | 4.01 | 2.70e-05 | 3.01 | 2.12e-05 | 3.03 | |
2 | 3.47e-03 | - | 2.23e-04 | - | 9.86e-03 | - | 8.27e-03 | - | |
4 | 2.24e-04 | 3.97 | 7.13e-06 | 4.99 | 6.63e-04 | 3.88 | 5.20e-04 | 4.02 | |
4 | 8 | 1.40e-05 | 4.12 | 2.30e-07 | 5.03 | 4.14e-05 | 4.00 | 3.30e-05 | 3.94 |
16 | 8.80e-07 | 3.98 | 7.07e-09 | 5.01 | 2.59e-06 | 4.08 | 2.07e-06 | 4.05 | |
32 | 5.50e-08 | 4.00 | 2.22e-10 | 5.02 | 1.62e-07 | 4.05 | 1.30e-07 | 4.04 |
error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||
4 | 4.24e-04 | - | 4.42e-04 | - | 1.45e-02 | - | 4.98e-01 | - | |
8 | 2.75e-05 | 3.94 | 2.94e-05 | 3.91 | 1.38e-03 | 3.39 | 8.75e-02 | 2.51 | |
3 | 16 | 1.75e-06 | 3.97 | 1.87e-06 | 3.98 | 1.03e-04 | 3.74 | 1.25e-02 | 2.81 |
32 | 1.10e-07 | 4.00 | 1.17e-07 | 3.99 | 6.85e-06 | 3.91 | 1.63e-03 | 2.94 | |
64 | 6.86e-09 | 4.00 | 7.34e-09 | 4.00 | 4.36e-07 | 3.97 | 2.05e-04 | 2.99 | |
Case 2 | |||||||||
4 | 3.39e-04 | - | 3.42e-04 | - | 1.37e-02 | - | 4.88e-01 | - | |
8 | 2.25e-05 | 3.92 | 2.69e-05 | 3.67 | 1.31e-03 | 3.39 | 8.56e-02 | 2.51 | |
3 | 16 | 1.44e-06 | 3.96 | 1.88e-06 | 3.84 | 9.74e-05 | 3.74 | 1.22e-02 | 2.81 |
32 | 9.03e-08 | 4.00 | 1.23e-07 | 3.93 | 6.46e-06 | 3.91 | 1.58e-03 | 2.95 | |
64 | 5.64e-09 | 4.00 | 7.87e-09 | 3.97 | 4.11e-07 | 3.98 | 2.00e-04 | 2.99 | |
Case 3 | |||||||||
4 | 3.36e-04 | - | 3.53e-04 | - | 1.37e-02 | - | 4.88e-01 | - | |
8 | 2.26e-05 | 3.89 | 2.82e-05 | 3.65 | 1.30e-03 | 3.39 | 8.56e-02 | 2.51 | |
3 | 16 | 1.44e-06 | 3.97 | 1.97e-06 | 3.84 | 9.72e-05 | 3.75 | 1.22e-02 | 2.81 |
32 | 9.05e-08 | 4.00 | 1.29e-07 | 3.93 | 6.45e-06 | 3.91 | 1.59e-03 | 2.95 | |
64 | 5.66e-09 | 4.00 | 8.26e-09 | 3.97 | 4.09e-07 | 3.98 | 2.00e-04 | 2.99 |
error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||
4 | 4.24e-04 | - | 4.42e-04 | - | 1.45e-02 | - | 4.98e-01 | - | |
8 | 2.75e-05 | 3.94 | 2.94e-05 | 3.91 | 1.38e-03 | 3.39 | 8.75e-02 | 2.51 | |
3 | 16 | 1.75e-06 | 3.97 | 1.87e-06 | 3.98 | 1.03e-04 | 3.74 | 1.25e-02 | 2.81 |
32 | 1.10e-07 | 4.00 | 1.17e-07 | 3.99 | 6.85e-06 | 3.91 | 1.63e-03 | 2.94 | |
64 | 6.86e-09 | 4.00 | 7.34e-09 | 4.00 | 4.36e-07 | 3.97 | 2.05e-04 | 2.99 | |
Case 2 | |||||||||
4 | 3.39e-04 | - | 3.42e-04 | - | 1.37e-02 | - | 4.88e-01 | - | |
8 | 2.25e-05 | 3.92 | 2.69e-05 | 3.67 | 1.31e-03 | 3.39 | 8.56e-02 | 2.51 | |
3 | 16 | 1.44e-06 | 3.96 | 1.88e-06 | 3.84 | 9.74e-05 | 3.74 | 1.22e-02 | 2.81 |
32 | 9.03e-08 | 4.00 | 1.23e-07 | 3.93 | 6.46e-06 | 3.91 | 1.58e-03 | 2.95 | |
64 | 5.64e-09 | 4.00 | 7.87e-09 | 3.97 | 4.11e-07 | 3.98 | 2.00e-04 | 2.99 | |
Case 3 | |||||||||
4 | 3.36e-04 | - | 3.53e-04 | - | 1.37e-02 | - | 4.88e-01 | - | |
8 | 2.26e-05 | 3.89 | 2.82e-05 | 3.65 | 1.30e-03 | 3.39 | 8.56e-02 | 2.51 | |
3 | 16 | 1.44e-06 | 3.97 | 1.97e-06 | 3.84 | 9.72e-05 | 3.75 | 1.22e-02 | 2.81 |
32 | 9.05e-08 | 4.00 | 1.29e-07 | 3.93 | 6.45e-06 | 3.91 | 1.59e-03 | 2.95 | |
64 | 5.66e-09 | 4.00 | 8.26e-09 | 3.97 | 4.09e-07 | 3.98 | 2.00e-04 | 2.99 |
error | order | error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||||
4 | 2.56e-06 | - | 1.71e-06 | - | 4.10e-05 | - | 1.19e-03 | - | 6.21e-02 | - | |
8 | 4.06e-08 | 5.98 | 3.46e-08 | 5.62 | 8.78e-07 | 5.55 | 4.83e-05 | 4.63 | 4.91e-03 | 3.66 | |
4 | 16 | 6.25e-10 | 6.02 | 6.31e-10 | 5.78 | 1.35e-08 | 6.03 | 1.38e-06 | 5.13 | 2.90e-04 | 4.08 |
32 | 1.05e-11 | 5.89 | 1.03e-11 | 5.94 | 2.25e-10 | 5.90 | 4.78e-08 | 4.85 | 1.94e-05 | 3.90 | |
64 | 1.63e-13 | 6.01 | 1.65e-13 | 5.96 | 4.02e-12 | 5.81 | 1.52e-09 | 4.98 | 1.24e-06 | 3.97 | |
Case 2 | |||||||||||
4 | 1.25e-06 | - | 8.07e-07 | - | 3.91e-05 | - | 1.16e-03 | - | 6.07e-02 | - | |
8 | 2.01e-08 | 5.96 | 2.10e-08 | 5.27 | 8.31e-07 | 5.56 | 4.62e-05 | 4.65 | 4.79e-03 | 3.66 | |
4 | 16 | 3.10e-10 | 6.02 | 3.99e-10 | 5.71 | 1.26e-08 | 6.05 | 1.33e-06 | 5.12 | 2.84e-04 | 4.08 |
32 | 5.12e-12 | 5.92 | 6.70e-12 | 5.90 | 2.13e-10 | 5.89 | 4.59e-08 | 4.86 | 1.89e-05 | 3.90 | |
64 | 8.02e-14 | 6.00 | 1.05e-13 | 6.00 | 3.89e-12 | 5.77 | 1.46e-09 | 4.98 | 1.21e-06 | 3.97 | |
Case 3 | |||||||||||
4 | 9.26e-07 | - | 5.75e-07 | - | 3.93e-05 | - | 1.16e-03 | - | 6.07e-02 | - | |
8 | 1.48e-08 | 5.96 | 1.54e-08 | 5.22 | 8.33e-07 | 5.56 | 4.62e-05 | 4.65 | 4.79e-03 | 3.66 | |
4 | 16 | 2.29e-10 | 6.02 | 2.95e-10 | 5.71 | 1.26e-08 | 6.05 | 1.33e-06 | 5.11 | 2.84e-04 | 4.08 |
32 | 3.81e-12 | 5.91 | 4.94e-12 | 5.90 | 2.13e-10 | 5.89 | 4.58e-08 | 4.86 | 1.89e-05 | 3.90 | |
64 | 5.74e-14 | 6.05 | 9.17e-14 | 5.75 | 3.91e-12 | 5.77 | 1.46e-09 | 4.98 | 1.21e-06 | 3.97 |
error | order | error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||||
4 | 2.56e-06 | - | 1.71e-06 | - | 4.10e-05 | - | 1.19e-03 | - | 6.21e-02 | - | |
8 | 4.06e-08 | 5.98 | 3.46e-08 | 5.62 | 8.78e-07 | 5.55 | 4.83e-05 | 4.63 | 4.91e-03 | 3.66 | |
4 | 16 | 6.25e-10 | 6.02 | 6.31e-10 | 5.78 | 1.35e-08 | 6.03 | 1.38e-06 | 5.13 | 2.90e-04 | 4.08 |
32 | 1.05e-11 | 5.89 | 1.03e-11 | 5.94 | 2.25e-10 | 5.90 | 4.78e-08 | 4.85 | 1.94e-05 | 3.90 | |
64 | 1.63e-13 | 6.01 | 1.65e-13 | 5.96 | 4.02e-12 | 5.81 | 1.52e-09 | 4.98 | 1.24e-06 | 3.97 | |
Case 2 | |||||||||||
4 | 1.25e-06 | - | 8.07e-07 | - | 3.91e-05 | - | 1.16e-03 | - | 6.07e-02 | - | |
8 | 2.01e-08 | 5.96 | 2.10e-08 | 5.27 | 8.31e-07 | 5.56 | 4.62e-05 | 4.65 | 4.79e-03 | 3.66 | |
4 | 16 | 3.10e-10 | 6.02 | 3.99e-10 | 5.71 | 1.26e-08 | 6.05 | 1.33e-06 | 5.12 | 2.84e-04 | 4.08 |
32 | 5.12e-12 | 5.92 | 6.70e-12 | 5.90 | 2.13e-10 | 5.89 | 4.59e-08 | 4.86 | 1.89e-05 | 3.90 | |
64 | 8.02e-14 | 6.00 | 1.05e-13 | 6.00 | 3.89e-12 | 5.77 | 1.46e-09 | 4.98 | 1.21e-06 | 3.97 | |
Case 3 | |||||||||||
4 | 9.26e-07 | - | 5.75e-07 | - | 3.93e-05 | - | 1.16e-03 | - | 6.07e-02 | - | |
8 | 1.48e-08 | 5.96 | 1.54e-08 | 5.22 | 8.33e-07 | 5.56 | 4.62e-05 | 4.65 | 4.79e-03 | 3.66 | |
4 | 16 | 2.29e-10 | 6.02 | 2.95e-10 | 5.71 | 1.26e-08 | 6.05 | 1.33e-06 | 5.11 | 2.84e-04 | 4.08 |
32 | 3.81e-12 | 5.91 | 4.94e-12 | 5.90 | 2.13e-10 | 5.89 | 4.58e-08 | 4.86 | 1.89e-05 | 3.90 | |
64 | 5.74e-14 | 6.05 | 9.17e-14 | 5.75 | 3.91e-12 | 5.77 | 1.46e-09 | 4.98 | 1.21e-06 | 3.97 |
error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||
4 | 2.12e-03 | - | 3.30e-03 | - | 1.29e-02 | - | 7.15e-02 | - | |
8 | 1.43e-04 | 3.89 | 5.10e-04 | 2.69 | 1.33e-03 | 3.28 | 1.35e-02 | 2.41 | |
3 | 16 | 8.87e-06 | 4.01 | 4.71e-05 | 3.44 | 1.01e-04 | 3.72 | 2.03e-03 | 2.73 |
32 | 5.49e-07 | 4.02 | 3.42e-06 | 3.78 | 6.67e-06 | 3.92 | 2.76e-04 | 2.88 | |
64 | 3.42e-08 | 4.00 | 2.24e-07 | 3.93 | 4.22e-07 | 3.98 | 3.59e-05 | 2.94 | |
Case 2 | |||||||||
4 | 2.30e-03 | - | 2.97e-03 | - | 1.42e-02 | - | 9.18e-02 | - | |
8 | 1.56e-04 | 3.89 | 5.01e-04 | 2.57 | 1.45e-03 | 3.30 | 1.71e-02 | 2.43 | |
3 | 16 | 9.62e-06 | 4.02 | 4.72e-05 | 3.41 | 1.09e-04 | 3.72 | 2.55e-03 | 2.74 |
32 | 5.95e-07 | 4.01 | 3.45e-06 | 3.77 | 7.25e-06 | 3.92 | 3.46e-04 | 2.88 | |
64 | 3.71e-08 | 4.00 | 2.26e-07 | 3.93 | 4.59e-07 | 3.98 | 4.49e-05 | 2.94 | |
Case 3 | |||||||||
4 | 2.37e-03 | - | 2.84e-03 | - | 1.45e-02 | - | 9.13e-02 | - | |
8 | 1.59e-04 | 3.89 | 4.85e-04 | 2.55 | 1.47e-03 | 3.30 | 1.70e-02 | 2.43 | |
3 | 16 | 9.82e-06 | 4.02 | 4.60e-05 | 3.40 | 1.11e-04 | 3.73 | 2.54e-03 | 2.74 |
32 | 6.08e-07 | 4.01 | 3.37e-06 | 3.77 | 7.33e-06 | 3.92 | 3.45e-04 | 2.88 | |
64 | 3.79e-08 | 4.00 | 2.21e-07 | 3.93 | 4.64e-07 | 3.98 | 4.49e-05 | 2.94 |
error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||
4 | 2.12e-03 | - | 3.30e-03 | - | 1.29e-02 | - | 7.15e-02 | - | |
8 | 1.43e-04 | 3.89 | 5.10e-04 | 2.69 | 1.33e-03 | 3.28 | 1.35e-02 | 2.41 | |
3 | 16 | 8.87e-06 | 4.01 | 4.71e-05 | 3.44 | 1.01e-04 | 3.72 | 2.03e-03 | 2.73 |
32 | 5.49e-07 | 4.02 | 3.42e-06 | 3.78 | 6.67e-06 | 3.92 | 2.76e-04 | 2.88 | |
64 | 3.42e-08 | 4.00 | 2.24e-07 | 3.93 | 4.22e-07 | 3.98 | 3.59e-05 | 2.94 | |
Case 2 | |||||||||
4 | 2.30e-03 | - | 2.97e-03 | - | 1.42e-02 | - | 9.18e-02 | - | |
8 | 1.56e-04 | 3.89 | 5.01e-04 | 2.57 | 1.45e-03 | 3.30 | 1.71e-02 | 2.43 | |
3 | 16 | 9.62e-06 | 4.02 | 4.72e-05 | 3.41 | 1.09e-04 | 3.72 | 2.55e-03 | 2.74 |
32 | 5.95e-07 | 4.01 | 3.45e-06 | 3.77 | 7.25e-06 | 3.92 | 3.46e-04 | 2.88 | |
64 | 3.71e-08 | 4.00 | 2.26e-07 | 3.93 | 4.59e-07 | 3.98 | 4.49e-05 | 2.94 | |
Case 3 | |||||||||
4 | 2.37e-03 | - | 2.84e-03 | - | 1.45e-02 | - | 9.13e-02 | - | |
8 | 1.59e-04 | 3.89 | 4.85e-04 | 2.55 | 1.47e-03 | 3.30 | 1.70e-02 | 2.43 | |
3 | 16 | 9.82e-06 | 4.02 | 4.60e-05 | 3.40 | 1.11e-04 | 3.73 | 2.54e-03 | 2.74 |
32 | 6.08e-07 | 4.01 | 3.37e-06 | 3.77 | 7.33e-06 | 3.92 | 3.45e-04 | 2.88 | |
64 | 3.79e-08 | 4.00 | 2.21e-07 | 3.93 | 4.64e-07 | 3.98 | 4.49e-05 | 2.94 |
error | order | error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||||
4 | 1.45e-05 | - | 1.16e-04 | - | 8.66e-05 | - | 1.00e-03 | - | 8.32e-03 | - | |
8 | 4.69e-07 | 4.95 | 3.01e-06 | 5.27 | 1.53e-06 | 5.82 | 3.87e-05 | 4.69 | 7.68e-04 | 3.44 | |
4 | 16 | 1.25e-08 | 5.23 | 4.84e-08 | 5.96 | 1.64e-08 | 6.55 | 1.14e-06 | 5.09 | 5.70e-05 | 3.75 |
32 | 2.23e-10 | 5.81 | 7.53e-10 | 6.01 | 4.01e-10 | 5.35 | 3.76e-08 | 4.92 | 3.73e-06 | 3.94 | |
64 | 3.61e-12 | 5.95 | 1.18e-11 | 6.00 | 7.73e-12 | 5.70 | 1.18e-09 | 4.99 | 2.34e-07 | 4.00 | |
Case 2 | |||||||||||
4 | 1.60e-05 | - | 1.15e-04 | - | 9.17e-05 | - | 1.09e-03 | - | 1.06e-02 | - | |
8 | 4.82e-07 | 5.05 | 3.09e-06 | 5.22 | 1.63e-06 | 5.81 | 4.18e-05 | 4.70 | 9.85e-04 | 3.42 | |
4 | 16 | 1.31e-08 | 5.20 | 5.06e-08 | 5.93 | 1.78e-08 | 6.52 | 1.25e-06 | 5.06 | 7.21e-05 | 3.77 |
32 | 2.35e-10 | 5.80 | 7.92e-10 | 6.00 | 4.08e-10 | 5.45 | 4.10e-08 | 4.93 | 4.68e-06 | 3.94 | |
64 | 3.80e-12 | 5.95 | 1.24e-11 | 6.00 | 7.92e-12 | 5.69 | 1.28e-09 | 5.00 | 2.93e-07 | 4.00 | |
Case 3 | |||||||||||
4 | 1.61e-05 | - | 1.15e-04 | - | 9.18e-05 | - | 1.09e-03 | - | 1.05e-02 | - | |
8 | 4.84e-07 | 5.06 | 3.08e-06 | 5.22 | 1.63e-06 | 5.82 | 4.18e-05 | 4.70 | 9.86e-04 | 3.42 | |
4 | 16 | 1.32e-08 | 5.20 | 5.07e-08 | 5.92 | 1.78e-08 | 6.52 | 1.25e-06 | 5.06 | 7.21e-05 | 3.77 |
32 | 2.37e-10 | 5.80 | 7.96e-10 | 5.99 | 4.06e-10 | 5.45 | 4.10e-08 | 4.93 | 4.68e-06 | 3.95 | |
64 | 3.84e-12 | 5.95 | 1.25e-11 | 6.00 | 7.89e-12 | 5.69 | 1.28e-09 | 5.00 | 2.93e-07 | 4.00 |
error | order | error | order | error | order | error | order | error | order | ||
Case 1 | |||||||||||
4 | 1.45e-05 | - | 1.16e-04 | - | 8.66e-05 | - | 1.00e-03 | - | 8.32e-03 | - | |
8 | 4.69e-07 | 4.95 | 3.01e-06 | 5.27 | 1.53e-06 | 5.82 | 3.87e-05 | 4.69 | 7.68e-04 | 3.44 | |
4 | 16 | 1.25e-08 | 5.23 | 4.84e-08 | 5.96 | 1.64e-08 | 6.55 | 1.14e-06 | 5.09 | 5.70e-05 | 3.75 |
32 | 2.23e-10 | 5.81 | 7.53e-10 | 6.01 | 4.01e-10 | 5.35 | 3.76e-08 | 4.92 | 3.73e-06 | 3.94 | |
64 | 3.61e-12 | 5.95 | 1.18e-11 | 6.00 | 7.73e-12 | 5.70 | 1.18e-09 | 4.99 | 2.34e-07 | 4.00 | |
Case 2 | |||||||||||
4 | 1.60e-05 | - | 1.15e-04 | - | 9.17e-05 | - | 1.09e-03 | - | 1.06e-02 | - | |
8 | 4.82e-07 | 5.05 | 3.09e-06 | 5.22 | 1.63e-06 | 5.81 | 4.18e-05 | 4.70 | 9.85e-04 | 3.42 | |
4 | 16 | 1.31e-08 | 5.20 | 5.06e-08 | 5.93 | 1.78e-08 | 6.52 | 1.25e-06 | 5.06 | 7.21e-05 | 3.77 |
32 | 2.35e-10 | 5.80 | 7.92e-10 | 6.00 | 4.08e-10 | 5.45 | 4.10e-08 | 4.93 | 4.68e-06 | 3.94 | |
64 | 3.80e-12 | 5.95 | 1.24e-11 | 6.00 | 7.92e-12 | 5.69 | 1.28e-09 | 5.00 | 2.93e-07 | 4.00 | |
Case 3 | |||||||||||
4 | 1.61e-05 | - | 1.15e-04 | - | 9.18e-05 | - | 1.09e-03 | - | 1.05e-02 | - | |
8 | 4.84e-07 | 5.06 | 3.08e-06 | 5.22 | 1.63e-06 | 5.82 | 4.18e-05 | 4.70 | 9.86e-04 | 3.42 | |
4 | 16 | 1.32e-08 | 5.20 | 5.07e-08 | 5.92 | 1.78e-08 | 6.52 | 1.25e-06 | 5.06 | 7.21e-05 | 3.77 |
32 | 2.37e-10 | 5.80 | 7.96e-10 | 5.99 | 4.06e-10 | 5.45 | 4.10e-08 | 4.93 | 4.68e-06 | 3.95 | |
64 | 3.84e-12 | 5.95 | 1.25e-11 | 6.00 | 7.89e-12 | 5.69 | 1.28e-09 | 5.00 | 2.93e-07 | 4.00 |
error | order | error | order | error | order | |||
Case 1 | Case 2 | Case 3 | ||||||
4 | 2.35e-02 | - | 1.89e-02 | - | 1.89e-02 | - | ||
8 | 3.46e-03 | 2.77 | 2.82e-03 | 2.75 | 2.82e-03 | 2.75 | ||
3 | 16 | 4.49e-04 | 2.94 | 3.67e-04 | 2.94 | 3.67e-04 | 2.94 | |
32 | 5.66e-05 | 2.99 | 4.64e-05 | 2.99 | 4.64e-05 | 2.99 | ||
64 | 7.10e-06 | 3.00 | 5.81e-06 | 3.00 | 5.81e-06 | 3.00 | ||
4 | 1.86e-01 | - | 1.93e-01 | - | 1.93e-01 | - | ||
8 | 2.65e-02 | 2.81 | 2.75e-02 | 2.81 | 2.75e-02 | 2.81 | ||
3 | 16 | 3.37e-03 | 2.97 | 3.51e-03 | 2.97 | 3.51e-03 | 2.97 | |
32 | 4.22e-04 | 3.00 | 4.39e-04 | 3.00 | 4.39e-04 | 3.00 | ||
64 | 5.27e-05 | 3.00 | 5.49e-05 | 3.00 | 5.49e-05 | 3.00 |
error | order | error | order | error | order | |||
Case 1 | Case 2 | Case 3 | ||||||
4 | 2.35e-02 | - | 1.89e-02 | - | 1.89e-02 | - | ||
8 | 3.46e-03 | 2.77 | 2.82e-03 | 2.75 | 2.82e-03 | 2.75 | ||
3 | 16 | 4.49e-04 | 2.94 | 3.67e-04 | 2.94 | 3.67e-04 | 2.94 | |
32 | 5.66e-05 | 2.99 | 4.64e-05 | 2.99 | 4.64e-05 | 2.99 | ||
64 | 7.10e-06 | 3.00 | 5.81e-06 | 3.00 | 5.81e-06 | 3.00 | ||
4 | 1.86e-01 | - | 1.93e-01 | - | 1.93e-01 | - | ||
8 | 2.65e-02 | 2.81 | 2.75e-02 | 2.81 | 2.75e-02 | 2.81 | ||
3 | 16 | 3.37e-03 | 2.97 | 3.51e-03 | 2.97 | 3.51e-03 | 2.97 | |
32 | 4.22e-04 | 3.00 | 4.39e-04 | 3.00 | 4.39e-04 | 3.00 | ||
64 | 5.27e-05 | 3.00 | 5.49e-05 | 3.00 | 5.49e-05 | 3.00 |
error | order | error | order | error | order | |||
Case 1 | Case 2 | Case 3 | ||||||
4 | 5.03e-03 | - | 4.48e-03 | - | 4.48e-03 | - | ||
8 | 3.53e-04 | 3.83 | 3.17e-04 | 3.82 | 3.17e-04 | 3.82 | ||
4 | 16 | 2.24e-05 | 3.98 | 2.01e-05 | 3.98 | 2.01e-05 | 3.98 | |
32 | 1.40e-06 | 4.00 | 1.26e-06 | 4.00 | 1.26e-06 | 4.00 | ||
64 | 8.75e-08 | 4.00 | 7.86e-08 | 4.00 | 7.86e-08 | 4.00 | ||
4 | 2.09e-02 | - | 2.18e-02 | - | 2.18e-02 | - | ||
8 | 1.32e-03 | 3.99 | 1.38e-03 | 3.99 | 1.38e-03 | 3.99 | ||
4 | 16 | 7.74e-05 | 4.09 | 8.12e-05 | 4.08 | 8.12e-05 | 4.08 | |
32 | 4.86e-06 | 3.99 | 5.09e-06 | 4.00 | 5.09e-06 | 4.00 | ||
64 | 3.06e-07 | 3.99 | 3.20e-07 | 3.99 | 3.20e-07 | 3.99 |
error | order | error | order | error | order | |||
Case 1 | Case 2 | Case 3 | ||||||
4 | 5.03e-03 | - | 4.48e-03 | - | 4.48e-03 | - | ||
8 | 3.53e-04 | 3.83 | 3.17e-04 | 3.82 | 3.17e-04 | 3.82 | ||
4 | 16 | 2.24e-05 | 3.98 | 2.01e-05 | 3.98 | 2.01e-05 | 3.98 | |
32 | 1.40e-06 | 4.00 | 1.26e-06 | 4.00 | 1.26e-06 | 4.00 | ||
64 | 8.75e-08 | 4.00 | 7.86e-08 | 4.00 | 7.86e-08 | 4.00 | ||
4 | 2.09e-02 | - | 2.18e-02 | - | 2.18e-02 | - | ||
8 | 1.32e-03 | 3.99 | 1.38e-03 | 3.99 | 1.38e-03 | 3.99 | ||
4 | 16 | 7.74e-05 | 4.09 | 8.12e-05 | 4.08 | 8.12e-05 | 4.08 | |
32 | 4.86e-06 | 3.99 | 5.09e-06 | 4.00 | 5.09e-06 | 4.00 | ||
64 | 3.06e-07 | 3.99 | 3.20e-07 | 3.99 | 3.20e-07 | 3.99 |
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