Solution Fields of Nonlinear Equations and Continuation Methods

Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ , where $F:R^m \to R^n $ is a given mapping and $m > n$. Such problems arise, for instance, in structural mechanics and then usually $m - n$ of the variables $x_i $ are designated as parameters. For the case $m = n + 1$ an existence theory for the regular curves of the solution field is developed here. Then approximate solutions are considered and shown to be solutions of certain perturbed problems. These results are used to prove that for the continuation methods with Eider-predictor and Newton-corrector a particular steplength algorithm is guaranteed to trace any regular solution of the field. Some numerical aspects of the procedure are discussed and a numerical example is included to illustrate the effectiveness of the approach.