Abstract
In this paper, by the use of the residual vector and an approximation to the steepest descent direction of the norm function, we develop a norm descent spectral method for solving symmetric nonlinear equations. The method based on the nomonotone line search techniques is showed to be globally convergent. A specific implementation of the method is given which exploits the recently developed cyclic Barzilai–Borwein (CBB) for unconstrained optimization. Preliminary numerical results indicate that the method is promising.
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Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)
Birgin, E.G., Krejic, N.K., Martínez, J.M.: Globally convergent inexact quasi-Newton methods for solving nonlinear systems. Numer. Algorithms 32, 249–260 (2003)
Bellavia, S., Morini, B.: A globally convergent Newton-GMRES subspace method for systems of nonlinear equations. SIAM J. Sci. Comput. 23, 940–960 (2001)
Brown, P.N., Saad, Y.: Convergence theory of nonlinear Newton-Krylov algorithms. SIAM J. Optim. 4, 297–330 (1994)
Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965)
Cheng, W.Y.: A two-term PRP based-descent method. Numer. Funct. Anal. Optim. 28, 1217–1230 (2007)
Cheng, W.Y., Xiao, Y.H., Hu, Q.J.: A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. J. Comput. Appl. Math. 224, 11–19 (2009)
Cheng, W.Y.: A PRP type method for systems of nonlienar equations. Math. Comput. Model. 50, 15–20 (2009)
La Cruz, W., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems: theory and experiments. Technical Report RT-04-08, Dpto. de Computation, UCV (2004)
La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583–599 (2003)
La Cruz, W., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006)
Dai, Y.H., Zhang, H.: An adaptive two-point stepsize gradient algorithm. Numer. Algorithms 27, 377–385 (2001)
Dai, Y.H., W.W. hager, Schittkowski, K., Zhang, H.: The cyclic Barzilar-Borwein method for unconstrained optimization. IMA J. Numer. Anal. 26, 604–627 (2006)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Dennis, J.E. Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, NJ (1983)
Dennis, J.E., Moré, J.J.: Quasi-Newton methods, motivation and theory. SIAM Rev. 19, 46–89 (1977)
Gu, G.Z., Li, D.H., Qi, L.Q., Zhou, S.Z.: Desecent direction of quasi-Newton methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 40, 1763–1774 (2002)
Gasparo, M.: A nonmonotone hybrid method for nonlinear systems. Optim. Methods Softw. 13, 79–94 (2000)
Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)
Griewank, A.: The ‘global’ convergence of Broyden-like methods with suitable line search. J. Aust. Math. Soc. Series B 28, 75–92 (1986)
Gilbert, J.C., Nocedal, J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2, 21–42 (1992)
Li, D.H., Cheng, W.Y.: Recent progress in the global convergence of quasi-Newton methods for nonlinear equations. Hokkaido Math. J. 36, 729–743 (2007)
Li, D.H., Fukushima, M.: A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optim. Methods Softw. 13, 583–599 (2000)
Li, D.H., Fukushima, M.: A globally and superlinearly convergent Gauss-Newton-based BFGS methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152–172 (1999)
Li, D.H., Fukushima, M.: A modified BFGS method and its global convergence in nonconvex minimization. J. Comput. Appl. Math. 129, 15–35 (2001)
Martínez, J.M.: A family of quasi-Newton methods for nonlinear equations with direct secant updates of matrix factorizations. SIAM J. Numer. Anal. 27, 1034–1049 (1990)
Martínez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124 97–121 (2000)
Polak, E., Ribière, G.: Note surla convergence des mèthodes de diretions conjuguèes. Rev. Francaise Imformat Recherche Opertionelle 16, 35–43 (1969)
Polak, B.T.: The conjugate gradient method in extreme problems. USSR Comput. Math. Math. Phys. 9, 94–112 (1969)
Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 355–369. Kluwer, Norwell, MA (1999)
Zhang, L., Zhou, W.J., Li, D.H.: A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629–640 (2006)
Zhang, L., Zhou, W.J.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196, 478–484 (2006)
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Supported by the NSF of China via grant 11071087, 11101081 and by Foundation for Distinguished Young Talents in Higher Education of Guangdong, China LYM10127 and the NSF of Dongguan University of Technology via grant ZN100024.
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Cheng, W., Chen, Z. Nonmonotone spectral method for large-scale symmetric nonlinear equations. Numer Algor 62, 149–162 (2013). https://doi.org/10.1007/s11075-012-9572-z
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DOI: https://doi.org/10.1007/s11075-012-9572-z