Quasi-Newton Methods for Discretized Non-linear Boundary Problems

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Abstract
The discretization of non-linear boundary problems generally leads to a finite system of non-linear algebraic equations, and it is to be expected that this latter has special structure arising both from the boundary problem and the method of discretization used. The numerical solution of the algebraic system represents a serious numerical problem, and it is the point of this paper to indicate that, in certain important cases, special purpose quasi-Newton methods can be constructed. We illustrate by considering a single nonlinear differential equation discretized by collocation and present experimental results which indicate that an improvement in performance can be expected from the special methods.

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