### Description

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.

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.

Status | Finished |
---|---|

Effective start/end date | 1/05/00 → 31/12/02 |

### Keywords

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