DOI: 10.13763/j.cnki.jhebnu.nse.2015.02.003

Inexact Newton-least Squares Methods for Singular Nonlinear System of Equations
YANG Jialing, CAO Dexin
College of Sciences, China University of Mining and Technology, Jiangsu Xuzhou 221116, China
Abstract:
Inexact methods of Newton's method constructed with Moore-Penrose inverse are given.First,inexact Gauss-Newton method and inexact Levenberg-Marquardt method are deduced by taking an approximate solution of the least squares solution of Newton equations.Second,the inexact method is constructed by taking an approximate matrix of Moore-Penrose inverse of Jacobian matrix,and its convergence is proved.Third,the inexact method is constructed by using local information instead of the whole information of the Jacobian matrix,and the convergence is proved.The numerical example also indicates its superiority in solving large system of equations.

[1]KANTOROVICH L V,AKILOV G P.Functional Analysis[M].Oxford: Pergamon,1982.
[2]SMALE S.Newton's Method Estimates from Data at One Point[M]//EWING K,GROSS K,MARTIN C.The Merging of Disciplines: New Directions in Pure,Applied and Computational Mathematics,New York: Springer,1986:185-196.
[3]TRAUB J F,WOZNIAKOWSKI H.Convergence and Complexity of Newton Iteration[J].J Assoc Comput Math,1979,29:250-258.
[4]WANG Xinhua.Convergence of Newton's Method[J].Chinese Science Bulletin,1980,25:36-37.
[5]ORTEGA J M,RHEINBOLDT W C.Iterative Solution of Nonlinear Equations in Several Variables[M].New York: Academic Press,1970.
[6]ANEREW.On Newton-iterative Methods for the Solution of Systems of Nonlinear Equations[J].SIAM J Nuner Anal,1978,15:755-771.
[7]DEMBO R S,EISENSTAT S C,STEIHAUG T.Inexact Newton Methods[J].SIAM J Nuner Anal,1982,14:400-408.
[8]DEMBO R S,STEIHAUG T.Truncated-Newton Algorithms for Large-scale Unconstrained Optimization[J].Math Programming,1983,26:190-212.
[9]LIU Hao,NI Qin.Incomplete Jacobian Newton Method for Nonlinear Equations[J].Computer and Mathematics with Applications,2008,56:218-227.
[10]陈飞,王海军,曹苏玉.求解非线性方程组的改进不精确雅可比牛顿法[J].计算机工程与应用,2014,14:45-47.
[11]REDDIN.On Newton's Method for Singular Problem[J].SIAM J Nuner Anal,1987,15:993-994.
[12]KELLEY.A New Acceleration Method for Newton's Method at Singular Points[J].SIAM J Nuner Anal,1983,20:1001-1009.
[13]DECKER D W,KELLEY C T.Convergence Acceleration for Newton's Method at Singular Point[J].SIAM J Nuner Anal,1982,19:219-229.
[14]DECKER D W,KELLEY C T.Convergence Rates for Newton's Method at Singular Points[J].SIAM J Nuner Anal,1983,20:296-314.
[15]GRIEWANK A.On Solving Nonlinear Equations with Simple Singulartise or Nearly Singular Solutions[J].SIAM Rev,1985,27:537-563.
[16]ALLGOWER E L,BOHMER K.Resolving Singular Nonlinear Equations[J].Rocky Mount Math J,1987,18:225-268.
[17]吴国桢,王金华.关于奇异非线性方程组的Newton法的收敛性[J].浙江大学学报: 理学版,2008,35(1):27-31.
[18]PERIS R,MARQUINA A,CANDELA V.The Convergence of the Perturbed Newton Method and Its Application for Ill-conditioned Problems[J].Applied Mathematics and Computation,2011,218:2988-3001.
[19]BEN ISRALEL A,GREVILLE T N E.Generalized Inverse:Theory and Application[M].2th ed.New York: Spring-verlag,2002.
[20]CHEN J,LI W.Convergence of Gauss-Newton's Method and Uniqueness of the Solution[J].Applied Mathematics and Computation,2005,170:686-705.
[21]CHEN J.The Convergence Analysis of Inexact Gauss-Newton Methods for Nonlinear Problems[J].Comput Optim Appl,2008,40:97-118.
[22]WRIGHT S J,HOLT J N.An Inexact Levenberg-Marquardt Method for Large Sparse Nonlinear Least Squares[J].The Journal of the Australian Mathematical Society(Series B):Applied Mathematics,1985,26:387-403.
[23]HIROSHIGE D,NOBUO Y,MASSO F.Convergence Properties of the Inexact Levenberg-marquardt Method Under Local Error Bound Conditions[J].Optimization Methods and Software,2002,17:605-626.
[24]TOUTOUNIAN F,ATAEI A.A New Method for Computing Moore-penrose Inverse Matrices[J].Journal of Computational and Applied Mathematics,2009,228:412-417.
[25]CHEN H B,WANG Y J.A Family of Higher-order Convergent Iterative Methods for Computing the Moore-Penrose Inverse[J].Applied Mathematics and Computation,2011,218:4012-4016.