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A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems
Semi-local convergence of the Newton-HSS method under the center Lipschitz condition
1. | School of Science, Jiangnan University, Wuxi, 214122, P. R. China |
2. | Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China |
Newton-type methods have gained much attention in the past decades, especially for the semilocal convergence based on no information around the solution $x_*$ of the target nonlinear equation. For large sparse non-Hermitian positive definite systems of nonlinear equation, assuming that the nonlinear operator satisfies the center Lipschitz condition, which is wider than usual Lipschtiz condition and H$ö$lder continuous condition, we establish a new Newton-Kantorovich convergence theorem for the Newton-HSS method. Once the convergence criteria is satisfied, the iteration sequence $\{x_k\}_{k = 0}^∞$ generated by the Newton-HSS method is well defined, and converges to the solution $x_*$. Numerical results illustrate the effect.
References:
[1] |
H. An and Z. Bai,
A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57 (2007), 235-252.
doi: 10.1016/j.apnum.2006.02.007. |
[2] |
H. An, Z. Mo and X. Liu,
A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math., 20 (2007), 47-60.
doi: 10.1016/j.cam.2005.12.030. |
[3] |
O. Axelsson and G. F. Carey,
On the numerical solution of two-point singularly perturbed boundary value problems, Comput. Method Appl. Mech. Eng., 50 (1985), 217-229.
doi: 10.1016/0045-7825(85)90094-5. |
[4] |
O. Axelsson and M. Nikolova,
Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D, Computing, 61 (1998), 331-357.
doi: 10.1007/BF02684384. |
[5] |
Z. Bai, G. H. Golub and M. K. Ng,
Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626.
doi: 10.1137/S0895479801395458. |
[6] |
Z. Bai and X. Guo,
On Newton-HSS method for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math., 28 (2010), 235-260.
doi: 10.4208/jcm.2009.10-m2836. |
[7] |
S. Bellavia and B. Morini,
A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM J. Sci. Comput., 23 (2001), 940-960.
doi: 10.1137/S1064827599363976. |
[8] |
M. Chen, R. Lin and Q. Wu,
Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition, J. Comput. Appl. Math., 264 (2014), 115-130.
doi: 10.1016/j.cam.2013.12.047. |
[9] |
M. Chen, Q. Wu and R. Lin,
Semilocal convergence analysis for the Modified Newton-HSS method under the H$ö$lder condition, Numer. Algor., 72 (2016), 667-685.
doi: 10.1007/s11075-015-0061-z. |
[10] |
R. S. Dembo, S. C. Eisenstat and T. Steihaug,
Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408.
doi: 10.1137/0719025. |
[11] |
O. P. Ferreira and B. F. Svaiter,
Kantorovich's majorants principle for Newton's method, Comput. Optim. Appl., 42 (2009), 213-229.
doi: 10.1007/s10589-007-9082-4. |
[12] |
X. Guo,
On semilocal convergence of inexact Newton methods, J. Comput. Math., 25 (2007), 231-242.
|
[13] |
X. Guo,
On the convergence of Newton's method in Banach space, Journal of Zhejiang University(Sciences Edition), 27 (2000), 484-492.
|
[14] |
X. Guo and I. S. Duff,
Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl., 18 (2011), 299-315.
doi: 10.1002/nla.713. |
[15] |
L. V. Kantorovich and G. P. Akilov,
Functional Analysis in Normed Spaces, Oxford, Pergamon, 1964. |
[16] |
C. Li and K. F. NG,
Majorizing functions and convergence of the Gauss-Newton method for convex composite optimation, SIAM J. Optim., 18 (2007), 613-642.
doi: 10.1137/06065622X. |
[17] |
J. M. Ortega and W. C. Rheinbolt,
Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. |
[18] |
W. C. Rheinboldt,
Methods of Solving Systems of Nonlinear Equations, 2$^{nd}$ edition, SIAM, Philadelphia, 1998.
doi: 10.1137/1.9781611970012. |
[19] |
Y. Saad and M. Schultz,
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. Sci. Stat. Comput., 7 (1986), 856-869.
doi: 10.1137/0907058. |
[20] |
W. Shen and C. Li,
Kantorovich-type convergence criterion for inexact Newton methods, Appl. Numer. Math., 59 (2009), 1599-1611.
doi: 10.1016/j.apnum.2008.11.002. |
[21] |
X. Wang,
Convergence of Newton's method and inverse function theorem in Banach space, Math. Comput., 68 (1999), 169-185.
doi: 10.1090/S0025-5718-99-00999-0. |
[22] |
A. Yang and Y. Wu,
Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algebra, Control. Optim., 2 (2012), 839-853.
doi: 10.3934/naco.2012.2.839. |
[23] |
H. Zhong, G. Chen and X. Guo,
On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algor., 69 (2015), 553-567.
doi: 10.1007/s11075-014-9912-2. |
[24] |
H. Zhong, G. Wu and G. Chen,
A flexible and adaptive simpler block GMRES with deflated restarting for linear systems with multiple right-hand sides, J. Comput. Appl. Math., 282 (2015), 139-156.
doi: 10.1016/j.cam.2014.12.040. |
show all references
References:
[1] |
H. An and Z. Bai,
A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57 (2007), 235-252.
doi: 10.1016/j.apnum.2006.02.007. |
[2] |
H. An, Z. Mo and X. Liu,
A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math., 20 (2007), 47-60.
doi: 10.1016/j.cam.2005.12.030. |
[3] |
O. Axelsson and G. F. Carey,
On the numerical solution of two-point singularly perturbed boundary value problems, Comput. Method Appl. Mech. Eng., 50 (1985), 217-229.
doi: 10.1016/0045-7825(85)90094-5. |
[4] |
O. Axelsson and M. Nikolova,
Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D, Computing, 61 (1998), 331-357.
doi: 10.1007/BF02684384. |
[5] |
Z. Bai, G. H. Golub and M. K. Ng,
Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626.
doi: 10.1137/S0895479801395458. |
[6] |
Z. Bai and X. Guo,
On Newton-HSS method for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math., 28 (2010), 235-260.
doi: 10.4208/jcm.2009.10-m2836. |
[7] |
S. Bellavia and B. Morini,
A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM J. Sci. Comput., 23 (2001), 940-960.
doi: 10.1137/S1064827599363976. |
[8] |
M. Chen, R. Lin and Q. Wu,
Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition, J. Comput. Appl. Math., 264 (2014), 115-130.
doi: 10.1016/j.cam.2013.12.047. |
[9] |
M. Chen, Q. Wu and R. Lin,
Semilocal convergence analysis for the Modified Newton-HSS method under the H$ö$lder condition, Numer. Algor., 72 (2016), 667-685.
doi: 10.1007/s11075-015-0061-z. |
[10] |
R. S. Dembo, S. C. Eisenstat and T. Steihaug,
Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408.
doi: 10.1137/0719025. |
[11] |
O. P. Ferreira and B. F. Svaiter,
Kantorovich's majorants principle for Newton's method, Comput. Optim. Appl., 42 (2009), 213-229.
doi: 10.1007/s10589-007-9082-4. |
[12] |
X. Guo,
On semilocal convergence of inexact Newton methods, J. Comput. Math., 25 (2007), 231-242.
|
[13] |
X. Guo,
On the convergence of Newton's method in Banach space, Journal of Zhejiang University(Sciences Edition), 27 (2000), 484-492.
|
[14] |
X. Guo and I. S. Duff,
Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl., 18 (2011), 299-315.
doi: 10.1002/nla.713. |
[15] |
L. V. Kantorovich and G. P. Akilov,
Functional Analysis in Normed Spaces, Oxford, Pergamon, 1964. |
[16] |
C. Li and K. F. NG,
Majorizing functions and convergence of the Gauss-Newton method for convex composite optimation, SIAM J. Optim., 18 (2007), 613-642.
doi: 10.1137/06065622X. |
[17] |
J. M. Ortega and W. C. Rheinbolt,
Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. |
[18] |
W. C. Rheinboldt,
Methods of Solving Systems of Nonlinear Equations, 2$^{nd}$ edition, SIAM, Philadelphia, 1998.
doi: 10.1137/1.9781611970012. |
[19] |
Y. Saad and M. Schultz,
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. Sci. Stat. Comput., 7 (1986), 856-869.
doi: 10.1137/0907058. |
[20] |
W. Shen and C. Li,
Kantorovich-type convergence criterion for inexact Newton methods, Appl. Numer. Math., 59 (2009), 1599-1611.
doi: 10.1016/j.apnum.2008.11.002. |
[21] |
X. Wang,
Convergence of Newton's method and inverse function theorem in Banach space, Math. Comput., 68 (1999), 169-185.
doi: 10.1090/S0025-5718-99-00999-0. |
[22] |
A. Yang and Y. Wu,
Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algebra, Control. Optim., 2 (2012), 839-853.
doi: 10.3934/naco.2012.2.839. |
[23] |
H. Zhong, G. Chen and X. Guo,
On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algor., 69 (2015), 553-567.
doi: 10.1007/s11075-014-9912-2. |
[24] |
H. Zhong, G. Wu and G. Chen,
A flexible and adaptive simpler block GMRES with deflated restarting for linear systems with multiple right-hand sides, J. Comput. Appl. Math., 282 (2015), 139-156.
doi: 10.1016/j.cam.2014.12.040. |
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1 | 20.2504 | 1.1624 | 75.9105 | 0.9956 | 0.004 | 4925 | 5.0839e+04 |
2 | 20.2504 | 1.1542 | 151.8310 | 0.9936 | 0.006 | 5577 | 5.1564e+04 |
10 | 20.2504 | 0.9331 | 761.3715 | 0.9339 | 0.070 | 3322 | 7.8891e+04 |
20 | 20.2504 | 0.5542 | 1.5515e+03 | 0.7685 | 0.301 | 9798 | 2.2368e+05 |
|
|
|
|
|
|||
1 | 20.2504 | 1.1624 | 75.9105 | 0.9956 | 0.004 | 4925 | 5.0839e+04 |
2 | 20.2504 | 1.1542 | 151.8310 | 0.9936 | 0.006 | 5577 | 5.1564e+04 |
10 | 20.2504 | 0.9331 | 761.3715 | 0.9339 | 0.070 | 3322 | 7.8891e+04 |
20 | 20.2504 | 0.5542 | 1.5515e+03 | 0.7685 | 0.301 | 9798 | 2.2368e+05 |
|
|
|
|
|
|||
1 | 15.5883 | 1.9189 | 66.8682 | 0.9966 | 0.003 | 5242 | 5.7083e+04 |
2 | 15.5883 | 1.9062 | 133.7451 | 0.9951 | 0.004 | 2542 | 5.7845e+04 |
10 | 15.5883 | 1.5606 | 670.5120 | 0.9496 | 0.053 | 24923 | 8.6297e+04 |
20 | 15.5883 | 0.9485 | 1.3620e+03 | 0.8198 | 0.219 | 3142 | 2.3364e+05 |
|
|
|
|
|
|||
1 | 15.5883 | 1.9189 | 66.8682 | 0.9966 | 0.003 | 5242 | 5.7083e+04 |
2 | 15.5883 | 1.9062 | 133.7451 | 0.9951 | 0.004 | 2542 | 5.7845e+04 |
10 | 15.5883 | 1.5606 | 670.5120 | 0.9496 | 0.053 | 24923 | 8.6297e+04 |
20 | 15.5883 | 0.9485 | 1.3620e+03 | 0.8198 | 0.219 | 3142 | 2.3364e+05 |
|
|
|
|
|
|||
1 | 12.7405 | 2.8827 | 60.7730 | 0.9972 | 0.002 | 2886 | 6.0552e+04 |
2 | 12.7405 | 2.8643 | 121.5538 | 0.9960 | 0.003 | 2357 | 6.1334e+04 |
10 | 12.7405 | 2.3595 | 609.2850 | 0.9592 | 0.042 | 4785 | 9.0384e+04 |
20 | 12.7405 | 1.4504 | 1.2351e+03 | 0.8525 | 0.173 | 76858 | 2.3920e+05 |
|
|
|
|
|
|||
1 | 12.7405 | 2.8827 | 60.7730 | 0.9972 | 0.002 | 2886 | 6.0552e+04 |
2 | 12.7405 | 2.8643 | 121.5538 | 0.9960 | 0.003 | 2357 | 6.1334e+04 |
10 | 12.7405 | 2.3595 | 609.2850 | 0.9592 | 0.042 | 4785 | 9.0384e+04 |
20 | 12.7405 | 1.4504 | 1.2351e+03 | 0.8525 | 0.173 | 76858 | 2.3920e+05 |
|
|
|
|
|
|||
1 | 26.6689 | 0.9065 | 100.6620 | 0.9967 | 0.0030 | 6591 | 8.3595e+04 |
2 | 26.6689 | 0.9015 | 201.3317 | 0.9952 | 0.0040 | 2863 | 8.4514e+04 |
10 | 26.6689 | 0.7627 | 1.0083e+03 | 0.9500 | 0.0520 | 4083 | 1.1808e+05 |
20 | 26.6689 | 0.4943 | 2.0385e+03 | 0.8212 | 0.2170 | 3563 | 2.8113e+05 |
|
|
|
|
|
|||
1 | 26.6689 | 0.9065 | 100.6620 | 0.9967 | 0.0030 | 6591 | 8.3595e+04 |
2 | 26.6689 | 0.9015 | 201.3317 | 0.9952 | 0.0040 | 2863 | 8.4514e+04 |
10 | 26.6689 | 0.7627 | 1.0083e+03 | 0.9500 | 0.0520 | 4083 | 1.1808e+05 |
20 | 26.6689 | 0.4943 | 2.0385e+03 | 0.8212 | 0.2170 | 3563 | 2.8113e+05 |
|
|
|
|
|
|||
1 | 20.4520 | 1.4830 | 88.3329 | 0.9974 | 0.0020 | 3983 | 9.5568e+04 |
2 | 20.4520 | 1.4754 | 176.6724 | 0.9963 | 0.0030 | 3396 | 9.6556e+04 |
10 | 20.4520 | 1.2610 | 884.7154 | 0.9619 | 0.0390 | 4261 | 1.3219e+05 |
20 | 20.4520 | 0.8368 | 1.7854e+03 | 0.8617 | 0.1600 | 5757 | 3.0013e+05 |
|
|
|
|
|
|||
1 | 20.4520 | 1.4830 | 88.3329 | 0.9974 | 0.0020 | 3983 | 9.5568e+04 |
2 | 20.4520 | 1.4754 | 176.6724 | 0.9963 | 0.0030 | 3396 | 9.6556e+04 |
10 | 20.4520 | 1.2610 | 884.7154 | 0.9619 | 0.0390 | 4261 | 1.3219e+05 |
20 | 20.4520 | 0.8368 | 1.7854e+03 | 0.8617 | 0.1600 | 5757 | 3.0013e+05 |
|
|
|
|
|
|||
1 | 16.6546 | 2.2085 | 79.9039 | 0.9979 | 0.0020 | 23235 | 1.0317e+05 |
2 | 16.6546 | 2.1976 | 159.8137 | 0.9970 | 0.0020 | 2347 | 1.0419e+05 |
10 | 16.6546 | 1.8906 | 800.2221 | 0.9693 | 0.0310 | 3352 | 1.4081e+05 |
20 | 16.6546 | 1.2714 | 1.6131e+03 | 0.8873 | 0.1270 | 79278 | 3.1131e+05 |
|
|
|
|
|
|||
1 | 16.6546 | 2.2085 | 79.9039 | 0.9979 | 0.0020 | 23235 | 1.0317e+05 |
2 | 16.6546 | 2.1976 | 159.8137 | 0.9970 | 0.0020 | 2347 | 1.0419e+05 |
10 | 16.6546 | 1.8906 | 800.2221 | 0.9693 | 0.0310 | 3352 | 1.4081e+05 |
20 | 16.6546 | 1.2714 | 1.6131e+03 | 0.8873 | 0.1270 | 79278 | 3.1131e+05 |
Error estimates | CPU | Outer IT | |||||||||
|
|
|
|
|
| ||||||
1 | 6.15e-07 | 4.90e-07 | 4.56e-07 | 1.60 | 5.57 | 6.20 | 4 | 4 | 4 | ||
2 | 2.42e-07 | 1.63e-07 | 1.39e-07 | 1.849 | 2.98 | 5.73 | 5 | 5 | 5 | ||
10 | 3.64e-07 | 2.70e-07 | 1.88e-07 | 1.12 | 28.15 | 10.85 | 6 | 6 | 6 | ||
20 | 2.60e-07 | 2.31e-07 | 1.37e-07 | 3.14 | 3.83 | 162.82 | 7 | 7 | 7 |
Error estimates | CPU | Outer IT | |||||||||
|
|
|
|
|
| ||||||
1 | 6.15e-07 | 4.90e-07 | 4.56e-07 | 1.60 | 5.57 | 6.20 | 4 | 4 | 4 | ||
2 | 2.42e-07 | 1.63e-07 | 1.39e-07 | 1.849 | 2.98 | 5.73 | 5 | 5 | 5 | ||
10 | 3.64e-07 | 2.70e-07 | 1.88e-07 | 1.12 | 28.15 | 10.85 | 6 | 6 | 6 | ||
20 | 2.60e-07 | 2.31e-07 | 1.37e-07 | 3.14 | 3.83 | 162.82 | 7 | 7 | 7 |
Error estimates | CPU | Outer IT | |||||||||
|
|
|
|
|
| ||||||
1 | 4.02e-07 | 3.80e-07 | 3.38e-07 | 2.02 | 3.38 | 74.69 | 4 | 4 | 4 | ||
2 | 1.41e-07 | 1.48e-07 | 1.15e-07 | 0.93 | 2.95 | 8.30 | 5 | 5 | 5 | ||
10 | 3.47e-07 | 2.16e-07 | 2.54e-07 | 1.31 | 3.73 | 12.50 | 6 | 6 | 6 | ||
20 | 1.53e-07 | 1.33e-07 | 1.06e-07 | 1.15 | 4.94 | 260.67 | 7 | 7 | 7 |
Error estimates | CPU | Outer IT | |||||||||
|
|
|
|
|
| ||||||
1 | 4.02e-07 | 3.80e-07 | 3.38e-07 | 2.02 | 3.38 | 74.69 | 4 | 4 | 4 | ||
2 | 1.41e-07 | 1.48e-07 | 1.15e-07 | 0.93 | 2.95 | 8.30 | 5 | 5 | 5 | ||
10 | 3.47e-07 | 2.16e-07 | 2.54e-07 | 1.31 | 3.73 | 12.50 | 6 | 6 | 6 | ||
20 | 1.53e-07 | 1.33e-07 | 1.06e-07 | 1.15 | 4.94 | 260.67 | 7 | 7 | 7 |
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