Convergence study of methods for solving systems of nonlinear equations

The asymptotic convergence of parameterized variants of
Newton's method for the solution of nonlinear systems of
equations is considered in this project.
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
developed. We show that asymptotically, a
single linear system is solved per update of the scalar
parameter. It turns out that a 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.