On Steplength Algorithms for a Class of Continuation Methods

The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form Fx = b,F:D ⊂ Rn + 1 → Rⁿ, for given b ∈ Rⁿ. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third one is applicable to any corrector. Some numerical results show the effectiveness of the three algorithms. Finally some asymptotic analysis of continuation steps is given.