Globally Convergent Inexact-newton Method For Solving Reducible Nonlinear Systems Of Equations

A nonlinear system of equations is said to be reducible if it can be divided into m blocks (m>1) in such a way that the i-th block depends only on the first i blocks of unknowns. Different ways of handling the different blocks with the aim of solving the system have been proposed in the literature. When the dimension of the blocks is very large, it can be difficult to solve the linear Newtonian equations associated to them using direct solvers based on factorizations. In this case, the idea of using iterative linear solvers to deal with the blocks of the system separately is appealing. In this paper, a local convergence theory that justifies this procedure is presented. The theory also explains the behavior of a Block-Newton method under different types of perturbations. Moreover, a globally convergent modification of the basic Block Inexact-Newton algorithm is introduced so that, under suitable assumptions, convergence can be ensured, independently of the initial point considered.1311134Crandall, M.G., (1978) Nonlinear Evolution Equations, , Academic PressDean, E.J., (1985) A Model Trust Region Modification of Inexact Newton's Method for Nonlinear Two Point Boundary Value Problems, , TR No 85-6, Department of Mathematical Sciences, Rice University, Houston, TexasDembo, R.S., Eisenstat, S.C., Steihaug, T., Inexact Newton methods (1982) SIAM Journal on Numerical Analysis, 19, pp. 400-408Dennis, J.E., Martínez, J.M., Zhang, X., Triangular decomposition methods for solving reducible nonlinear systems of equations (1994) SIAM Journal on Optimization, 4, pp. 358-382Dennis, J.E., Schnabel, R.B., (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, , Prentice-Hall, Englewood CliffsEisenstat, S.C., Walker, H.F., Globally Convergent Inexact Newton Methods (1994) SIAM Journal on Optimization, 4, pp. 393-422Eisenstat, S.C., Walker, H.F., Choosing the forcing terms in an Inexact Newton method (1996) SIAM Journal on Scientific Computing, 17, pp. 16-32Feehery, W.F., Tolsma, J.E., Barton, P.I., Efficient sensitivity analysis of large - Scale differential - Algebraic systems (1997) Applied Numerical Mathematics, 25, pp. 41-54Lužanin, Z., Krejić, N., Herceg, D., Parameter selection for Inexact Newton method (1997) Nonlinear Analysis. Theory, Methods and Applications, 30, pp. 17-24Maly, T., Petzold, L.R., Numerical methods and software for sensitivity analysis of differential - Algebraic systems (1996) Applied Numerical Mathematics, 20, pp. 57-79Meis, T., Marcowitz, U., (1981) Numerical Solution of Partial Difference Equations, , Springer-Verlag, New York, Heidelberg, BerlinMyint, T.U., Debnath, L., (1987) Partial Differential Equations for Scientists and Engineers, , ElsevierOrtega, J.M., Rheinboldt, W.C., (1970) Iterative Solution of Nonlinear Equations in Several Variables, , Academic Press, N