| A major part of the execution time of a boundary value ordinary differential equation solver is in the computation of the Newton matrices which arise during the solution of difficult nonlinear problems. Various researchers have suggested using techniques which involve approximating the Newton matrix in order to reduce the costs associated with its construction. In this thesis, we investigate several previously suggested methods, and some generalizations of them. We implement several of these approximate methods which, in comparison to the exact Newton method, are shown to be less efficient overall. An alternative approach, suggested by our study of the approximate methods, involves adapting the computation of the exact Newton matrix to the specific Runge-Kutta scheme being used. Numerical results show that this leads to substantial computational savings. |
