zbMATH — the first resource for mathematics

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)

65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.