Inexact Quasi-Newton methods for sparse systems of nonlinear equations

In this paper we present the results obtained in 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 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 superlinearly too. The matrix H = [A + 1 ; : : : ; A + i ; : : : ; A + p ]; where A + i = A T i (A i A T i ) 1 is the Moore-Penrose pseudo inverse of the m i n block, A i is 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 131072 on the CRAY T3E under the MPI environment.