Inexact Quasi-Newton methods for sparse systems of nonlinear equations

In this paper, we present the results obtained by solving consistent sparse systems of n nonlinear equations F(x) = 0, by a Quasi-Newton method combined with a p block iterative row-projection linear solver of Cimmino type, 1 less than or equal to p << n. Under weak regularity conditions for F, it is proved that this Inexact Quasi-Newton method has a local, linear convergence in the energy norm induced by the preconditioned matrix HA, where A is an initial guess of the Jacobian matrix, and it may converge too superlinearly. The matrix H = [A(1)(+),...,A(i)(+),...,A(p)(+)], where A(i)(+) = A(i)(T)(A(i)A(i)(T))(-1) is the Moore-Penrose pseudo-inverse of the mi x n block A(i), the preconditioner. A simple partitioning of the Jacobian matrix was used for solving a set of nonlinear test problems with sizes ranging from 1024 to 131 072 on the CRAY T3E under the MPI environment