Convergence study of methods for solving systems of nonlinear equations

Project: Research


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.
Effective start/end date1/05/0031/12/02


  • componentwise Q-superlinear convergence.
  • Nonlinear systems of equations
  • homotopy-type method