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On a Steffensen’s type method and its behavior for semismooth equations. (English) Zbl 1096.65047
Summary: A generalization of Steffensen’s method is proposed. Our goal is to obtain similar convergence as Newton’s method, but without to evaluate any derivative and without to have stability problems. Convergence analysis and numerical results for semismooth equations are presented.

65H10Systems of nonlinear equations (numerical methods)
Full Text: DOI
[1] Amat, S.; Busquier, S.; Candela, V.: A class of quasi-Newton generalized Steffensen methods on Banach spaces. J. comp. Appl. math. 149, No. 2, 397-406 (2002) · Zbl 1016.65035
[2] Amat, S.; Busquier, S.: On a higher order secant method. Appl. math. Comp. 141, No. 2-3, 321-329 (2003) · Zbl 1035.65057
[3] Amat, S.; Busquier, S.: A modified secant method for semismooth equations. Appl. math. Lett. 16, No. 6, 877-881 (2003) · Zbl 1059.65042
[4] Potra, F. A.; Qi, L.; Sun, D.: Secant methods for semismooth equations. Numer. math. 80, 305-324 (1998) · Zbl 0914.65051
[5] Qi, L.; Sun, D.: A nonsmooth version of Newton’s method. Math. program. 58, 353-368 (1993) · Zbl 0780.90090
[6] Clarke, F. H.: Optimization and nonsmooth analysis. (1983) · Zbl 0582.49001
[7] Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. oper. Res. 18, 227-244 (1993) · Zbl 0776.65037
[8] Pang, J. S.; Qi, L.: Nonsmooth equations: motivation and algorithms. SIAM J. Optim. 3, 443-465 (1993) · Zbl 0784.90082
[9] Ortega, J. M.; Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. (1975) · Zbl 0949.65053
[10] Potra, F. A.: On Q-order and R-order of convergence. J. optim. Theory appl. 63, 415-431 (1989) · Zbl 0663.65049