Componentwise fast convergence in solving nonlinear equations
N. I. M. Gould, D. Orban, A. Sartenaer, Ph. L. Toint
Report 00-05
Newton's method for the solution of nonlinear systems of equations is
considered. The original system is perturbed by a term involving the variables
and a scalar parameter which is driven to ZERO as the iteration proceeds. The
exact local solutions to the perturbed systems then form a differentiable path
leading to a solution of the original system, the scalar parameter determining
the progress along the path. A homotopy-type algorithm, which involves an
inner iteration in which the perturbed systems are approximately solved, is
outlined. It is shown that asymptotically, a single linear system is solved
per update of the scalar parameter. It turns out that a {\em componentwise}
Q-superlinear rate may be attained under standard assumptions, and that this
rate may be made arbitrarily close to quadratic. Numerical experiments
illustrate the results and we discuss the relationships that this method
shares with interior methods in constrained optimization.