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Newton-like methods for solving underdetermined nonlinear equations with nondifferentiable terms. (English) Zbl 0823.65048
Authors’ abstract: We consider Newton-like methods for solving underdetermined systems of nonlinear equations with nondifferentiable terms. After presenting a local convergence analysis for the methods, we prove a semilocal convergence theorem as well as uniqueness of solution in a generalized sense. Another semilocal convergence theorem for the Newton-chord method is also established. Finally, a numerical example is given.
Reviewer: D.Braess (Bochum)

MSC:
65H10 Numerical computation of solutions to systems of equations
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[1] Allgower, E.L.; Georg, K., Numerical continuation methods, (1990), Springer Berlin · Zbl 0717.65030
[2] Argyros, I.K., On the solution of equations with nondifferentiable operators and the pták error estimates, Bit, 30, 752-754, (1990) · Zbl 0726.65063
[3] Argyros, I.K., On the convergence of generalized Newton methods and implicit functions, J. comput. appl. math., 43, 335-342, (1992) · Zbl 0787.65033
[4] Argyros, I.K.; Szidarovszky, F., The theory and applications of iteration methods, (1993), CRC Press Boca Raton, FL, Chapter 5 · Zbl 0802.65076
[5] Ben-Israel, A., A newton—raphson method for the solution of systems of equations, J. math. anal. appl., 15, 243-252, (1966) · Zbl 0139.10301
[6] Chen, X., On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms, Ann. inst. statist. math., 42, 387-401, (1990) · Zbl 0718.65039
[7] Chen, X.; Qi, L., A parameterized Newton method and a quasi-Newton method for nonsmooth equations, Comput. optim. appl., 3, 157-179, (1994) · Zbl 0821.65029
[8] Chen, X.; Yamamoto, T., Convergence domains of certain iterative methods for solving nonlinear equations, Numer. funct. anal. optim., 10, 37-48, (1989) · Zbl 0645.65028
[9] Chen, X.; Yamamoto, T., PCG methods applied to a system of nonlinear equations, J. comput. appl. math., 38, 61-75, (1991) · Zbl 0746.65045
[10] Chen, X.; Yamamoto, T., On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators, Computing, 49, 87-94, (1992) · Zbl 0801.65048
[11] Häuβler, W.M., A Kantorovich-type convergence analysis for the gauss—newton-method, Numer. math., 48, 119-125, (1986) · Zbl 0598.65025
[12] Heinkenschloß, M.; Kelley, C.T.; Tran, H.T., Fast algorithms for nonsmooth compact fixed point problems, SIAM J. numer. anal., 29, 1769-1792, (1992) · Zbl 0763.65040
[13] Lawson, C.L.; Hanson, R.J., Solving least square problems, (1974), Prentice-Hall Englewood Cliffs, NJ · Zbl 0185.40701
[14] Martínez, J.M., Quasi-Newton methods for solving underdetermined nonlinear simultaneous equations, J. comput. appl. math., 34, 171-190, (1991) · Zbl 0729.65035
[15] Meyn, K.H., Solution of underdetermined nonlinear equations by stationary iteration methods, Numer. math., 42, 161-172, (1983) · Zbl 0497.65026
[16] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1987), Academic Press New York · Zbl 0949.65053
[17] Walker, H.F., Newton-like methods for underdetermined systems, (), 679-689, Lectures in Appl. Math.
[18] Walker, H.F.; Watson, L.T., Least-change secant update methods for underdetermined systems, SIAM J. numer. anal., 27, 1227-1262, (1990) · Zbl 0733.65032
[19] Yamamoto, T., Uniqueness of the solution in a Kantorovich-type theorem of Häußler for the gauss—newton method, Japan J. appl. math., 6, 77-81, (1989) · Zbl 0667.65053
[20] Yamamoto, T.; Chen, X., Ball-convergence theorems and error estimates for certain iterative methods for nonlinear equations, Japan J. appl. math., 7, 131-143, (1990) · Zbl 0699.65042
[21] Zabrejko, P.P.; Nguen, D.F., The majorant method in the theory of newton—kantorovich approximations and the pták error estimates, Numer. funct. anal. optim., 9, 671-684, (1987) · Zbl 0627.65069
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