Abstract
The present paper is concerned with theoretical properties of the modified Newton-HSS method for large sparse non-Hermitian positive definite systems of nonlinear equations. Assuming that the nonlinear operator satisfies the Hölder continuity condition, a new semilocal convergence theorem for the modified Newton-HSS method is established. The Hölder continuity condition is milder than the usual Lipschitz condition. The semilocal convergence theorem is established by using the majorizing principle, which is based on the concept of majorizing sequence given by Kantorovich. Two real valued functions and two real sequences are used to establish the convergence criterion. Furthermore, a numerical example is given to show application of our theorem.
Similar content being viewed by others
References
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic, New York (1970)
Rheinboldt, W.C.: Methods of solving systems of nonlinear equations, The Second Edition. SIAM, Philadelphia (1998)
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton mehtods. SIAM J. Numer. Anal. 19, 400–408 (1982)
Saad, Y.: Iterative methods for sparse linear systems, The Second Edition. SIAM, Philadelphia (2003)
An, H.-B., Bai, Z.-Z.: A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. Appl. Numer. Math. 57, 235–252 (2007)
An, H.-B., Mo, Z.-Y., Liu, X.-P.: A choice of forcing terms in inexact Newton method. J. Comput. Appl. Math. 20, 47–60 (2007)
Knoll, D.A., Keyes, D.A.: Jacobian-free Newton-Krylov method: a survery of appproaches and applilcations. J. Comput. Phys. 193, 357–397 (2004)
Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)
Bai, Z.-Z., Golub, G.H., Ng, M.K.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428, 413–440 (2008)
Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)
Chan, L.C., Ng, M.K., Tsing, N.K.: Spectral analysis for HSS preconditioners. Numer. Math. Theor. Methods Appl. 1, 57–77 (2008)
Bai, Z.-Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76, 287–298 (2007)
Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844–863 (2005)
Bai, Z.-Z., Golub, G.H., Ng, M.K.: On successive overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14, 319–335 (2007)
Bai, Z.-Z., Golub, G.H., Li, C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28, 583–603 (2006)
Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1–23 (2007)
Benzi, M., Gander, M.J., Golub, G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT Numer. Math. 43, 881–900 (2003)
Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33, 343–369 (2013)
Bai, Z.-Z., Guo, X.-P.: The Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices. J. Comput. Math. 28, 235–260 (2010)
Wu, Q.-B., Chen, M.-H.: Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations. Numer. Algor. 64, 659–635 (2013)
Darvishi, M.T., Barati, A.: A third-order Newton-type method to solve systems of nonlinear equations. Appl. Math. Comput. 187, 630–635 (2007)
Bai, Z.-Z., Tong, P.-L.: On the affine invariant convergence theorems of inexact Newton method and Broyden’s method. J. UEST China 23, 535–540 (1994)
Zhou, J.-L., Zhang, S.-Y., Yang, G.-P., Tan, J.-R: A convergence theorem for the inexact Newton methods based on Hölder continuous Fréchet derivative. Appl. Math. Comput. 197, 206–211 (2008)
Guo, X.-P.: On semilocal convergence of inexact Neton methods. J. Comput. Math. 25, 231–242 (2007)
Wu, M.: A new semi-local convergence theorem for the inexact Newton methods. Appl. Math. Comput. 200, 80–86 (2008)
Shen, W.-P., Li, C.: Kantorovich-type convergence criterion for inexact Newton methods. Appl. Numer. Math. 59, 1599–1611 (2009)
Argyros, I.K.: On the semilocal convergence of inexact Newton methods in Banach spaces. J. Comput. Appl. Math. 228, 434–443 (2009)
Shen, W.-P., Li, C.: Convergence criterion of inexact methods for operators with Hölder continuous derivatives. Taiwanese J. Math. 12, 1865–1882 (2008)
Chen, M.-H., Lin, R.-F., Wu, Q.-B.: Convergence analysis of modified Newton-HSS method under Hölder continuous condition. J. Comput. Appl. Math. 264, 115–130 (2014)
Rheinboldt, W.C.: A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5, 42–63 (1968)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, M., Wu, Q. & Lin, R. Semilocal convergence analysis for the modified Newton-HSS method under the Hölder condition. Numer Algor 72, 667–685 (2016). https://doi.org/10.1007/s11075-015-0061-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-015-0061-z
Keywords
- Modified Newton-HlSS method
- Large sparse systems
- Nonlinear equations
- Hölder continuous condition
- Positive-definite Jacobian matrices
- Majorant principle