Partial Newton methods for a system of equations

Previous Article
Direct model reference adaptive control of linear systems with input/output delays
NACO Home
This Issue
Next Article
Introduction to the theory of splines with an optimal mesh. Linear Chebyshev splines and applications

2013, 3(3): 463-469. doi: 10.3934/naco.2013.3.463

Partial Newton methods for a system of equations

B. S. Goh ^1, , W. J. Leong ^2, and Z. Siri ^3,

1.	Curtin Sarawak Research Institute, Curtin University Sarawak, 98009 Miri, Sarawak, Malaysia
2.	Department of Mathematics, University Putra Malaysia, 43400 Serdang, Malaysia
3.	Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received December 2011 Revised May 2013 Published July 2013

Abstract
Related Papers

We define and analyse partial Newton iterations for the solutions of a system of algebraic equations. Firstly we focus on a linear system of equations which does not require a line search. To apply a partial Newton method to a system of nonlinear equations we need a line search to ensure that the linearized equations are valid approximations of the nonlinear equations. We also focus on the use of one or two components of the displacement vector to generate a convergent sequence. This approach is inspired by the Simplex Algorithm in Linear Programming. As expected the partial Newton iterations are found not to have the fast convergence properties of the full Newton method. But the proposed partial Newton iteration makes it significantly simpler and faster to compute in each iteration for a system of equations with many variables. This is because it uses only one or two variables instead of all the search variables in each iteration.

Keywords: partial Newton iterations, convergence., solution of equations, Newton method, subspace method.

Mathematics Subject Classification: Primary: 49M15, 65K10; Secondary: 90C30, 90C5.

Citation: B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463

References:

[1]	B. S. Goh, Greatest descent algorithms in unconstrained optimization,, J. Optim. Theory Appl., 142 (2009), 275. doi: 10.1007/s10957-009-9533-4. Google Scholar
[2]	B. S. Goh, Convergence of algorithms in optimization and solutions of nonlinear equations,, J. Optim. Theory Appl., 144 (2010), 43. doi: 10.1007/s10957-009-9583-7. Google Scholar
[3]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM Publication, (1995). doi: 10.1137/1.9781611970944. Google Scholar
[4]	J. P. LaSalle, "The Stability of Dynamical Systems,", SIAM Publication, (1976). doi: 10.1137/1.9781611970432. Google Scholar

show all references

References:

[1]	B. S. Goh, Greatest descent algorithms in unconstrained optimization,, J. Optim. Theory Appl., 142 (2009), 275. doi: 10.1007/s10957-009-9533-4. Google Scholar
[2]	B. S. Goh, Convergence of algorithms in optimization and solutions of nonlinear equations,, J. Optim. Theory Appl., 144 (2010), 43. doi: 10.1007/s10957-009-9583-7. Google Scholar
[3]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM Publication, (1995). doi: 10.1137/1.9781611970944. Google Scholar
[4]	J. P. LaSalle, "The Stability of Dynamical Systems,", SIAM Publication, (1976). doi: 10.1137/1.9781611970432. Google Scholar

[1]	Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008
[2]	Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial & Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235
[3]	Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007
[4]	T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201
[5]	Xiaojiao Tong, Felix F. Wu, Yongping Zhang, Zheng Yan, Yixin Ni. A semismooth Newton method for solving optimal power flow. Journal of Industrial & Management Optimization, 2007, 3 (3) : 553-567. doi: 10.3934/jimo.2007.3.553
[6]	Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems & Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677
[7]	Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial & Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003
[8]	Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
[9]	Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237
[10]	R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39
[11]	Hans J. Wolters. A Newton-type method for computing best segment approximations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 133-148. doi: 10.3934/cpaa.2004.3.133
[12]	Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143
[13]	Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[14]	Cheng-Dar Liou. Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method". Journal of Industrial & Management Optimization, 2012, 8 (3) : 727-732. doi: 10.3934/jimo.2012.8.727
[15]	Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems & Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353
[16]	Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1
[17]	Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial & Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247
[18]	Andy M. Yip, Wei Zhu. A fast modified Newton's method for curvature based denoising of 1D signals. Inverse Problems & Imaging, 2013, 7 (3) : 1075-1097. doi: 10.3934/ipi.2013.7.1075
[19]	Kuo-Hsiung Wang, Chuen-Wen Liao, Tseng-Chang Yen. Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method. Journal of Industrial & Management Optimization, 2010, 6 (1) : 197-207. doi: 10.3934/jimo.2010.6.197
[20]	Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018149

Impact Factor:

American Institute of Mathematical Sciences