A new trust-region method with line search for solving symmetric nonlinear equations

References

• Bertsekas , D. P. 1995 . Nonlinear Programming , Belmont, MA : Athena Scientific .
   Google Scholar
• Brown , P. N. and Saad , Y. 1994 . Convergence theory of nonlinear Newton–Kryloy algorithms . SIAM J. Optim , 4 : 297 – 330 .
   Web of Science ®Google Scholar
• Conn , A. R. , Gould , N. I.M. and Toint , P. L. 2000 . Trust Region Method , Philadelphia, PA : Society for Industrial and Applied Mathematics .
   Google Scholar
• Dennis , J. E. and Schnabel , R. B. 1983 . Numerical Methods for Unconstrained Optimization and Nonlinear Equations , Englewood Cliffs, NJ : Prentice-Hall .
   Google Scholar
• Dolan , E. D. and Moré , J. J. 2002 . Benchmarking optimization software with performance profiles . Math. Program. , 91 : 201 – 213 .
   Web of Science ®Google Scholar
• Fan , J. Y. 2003 . A modified Levenberg–Marquardt algorithm for singular system of nonlinear equations . J. Comput. Math. , 21 : 625 – 636 .
   Web of Science ®Google Scholar
• Fletcher , R. 1987 . Practical Methods of Optimizations , John Wiley & Sons, New York .
   Google Scholar
• Gomez-Ruggiero , M. , Martinez , J. M. and Moretti , A. 1992 . Comparing algorithms for solving sparse nonlinear systems of equations . SIAM J. Sci. Statist. Comput. , 23 : 459 – 483 .
   Web of Science ®Google Scholar
• Griewank , A. 1986 . The ‘global’ convergence of Broyden-like methods with a suitable line search . J. Aust. Math. Soc. Ser. B , 28 : 75 – 92 .
   Google Scholar
• Levenberg , K. 1944 . A method for the solution of certain nonlinear problem in least squares . Quart. Appl. Math. , 2 : 164 – 166 .
   Google Scholar
• Li , D. and Fukushima , M. 1999 . A global and superlinear convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations . SIAM J. Numer. Anal. , 37 : 152 – 172 .
   Web of Science ®Google Scholar
• Marquardt , D. W. 1963 . An algorithm for least-squares estimation of nonlinear inequalities . SIAM J. Appl. Math. , 11 : 431 – 441 .
   Web of Science ®Google Scholar
• Moré , J. J. , Garbow , B. S. and Hillström , K. E. 1981 . Testing unconstrained optimization software . ACM Trans. Math. Softw. , 7 : 17 – 41 .
   Web of Science ®Google Scholar
• Nocedal , J. and Wright , S. J. 1999 . Numerical Optimization , Springer, New York .
   Google Scholar
• Ortega , J. M. and Rheinboldt , W. C. 1970 . Iterative Solution of Nonlinear Equations in Several Variables , Academic Press, New York .
   Google Scholar
• Raydan , M. 1997 . The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem . SIAM J. Optim. , 7 : 26 – 33 .
   Web of Science ®Google Scholar
• Wang , Y. J. and Xiu , N. H. 2004 . Theory and Algorithms for Nonlinear Programming , Shanxi Science and Technology Press, Xian .
   Google Scholar
• Wei , Z. , Yuan , G. and Lian , Z. 2004 . An approximate Gauss–Newton-based BFGS method for solving symmetric nonlinear equations . Guangxi Sci , 11 ( 2 ) : 91 – 99 .
   Google Scholar
• Yamakawa , E. and Fukushima , M. 1999 . Testing parallel bariable transformation . Comput. Optim. Appl. , 13 : 253 – 274 .
   Web of Science ®Google Scholar
• Yamashita , N. and Fukushima , M. 2001 . On the rate of convergence of the Levenberg–Marquardt method . Computing , 15 : 239 – 249 .
   Google Scholar
• Yuan , Y. 1998 . Trust region algorithm for nonlinear equations . Information , 1 : 7 – 21 .
   Google Scholar
• Yuan , G. L. and Li , X. R. 2004 . An approximate Gauss–Newton-based BFGS method with descent directions for solving symmetric nonlinear equations . OR Trans , 8 ( 4 ) : 10 – 26 .
   Google Scholar
• Yuan , G. L. and Li , X. R. 2010 . A rank-one fitting method for solving symmetric nonlinear equations . J. Appl. Funct. Anal. , 5 : 389 – 407 .
   Google Scholar
• Yuan , G. L. and Lu , X. W. 2008 . A new backtracking inexact BFGS method for symmetric nonlinear equations . Comput. Math. Appl. , 55 : 116 – 129 .
   Web of Science ®Google Scholar
• Yuan , Y. and Sun , W. 1997 . Optimization Theory and Algorithm , Beijing : Scientific Publisher House .
   Google Scholar
• Yuan , G. L. , Lu , X. W. and Wei , Z. X. 2009 . BFGS trust-region method for symmetric nonlinear equations . J. Comput. Appl. Math. , 230 : 44 – 58 .
   Web of Science ®Google Scholar
• Yuan , G. L. , Chen , C. L. and Wei , Z. X. 2010 . A nonmonotone adaptive trust-region algorithm for symmetric nonlinear equations . Nat. Sci , 2 ( 4 ) : 373 – 378 .
   Google Scholar
• Zhang , J. and Wang , Y. 2003 . A new trust region method for nonlinear equations . Math. Methods Oper. Res. , 58 : 283 – 298 .
   Web of Science ®Google Scholar
• Zhu , D. 2005 . Nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations . Appl. Math. Comput. , 161 : 875 – 895 .
   Web of Science ®Google Scholar