  Pseudo-arclength continuation is a well-established method for generating a
numerical curve approximating the solution of an underdetermined system of
nonlinear equations. It is an inherently sequential predictor-corrector method
in which new approximate solutions are extrapolated from previously converged
results and then iteratively refined. Convergence of the iterative corrections
is guaranteed only for sufficiently small prediction steps. In high-dimensional
systems, corrector steps are extremely costly to compute and the prediction
step-length must be adapted carefully to avoid failed steps or unnecessarily
slow progress. We describe a parallel method for adapting the step-length
employing several predictor-corrector sequences of different step lengths
computed concurrently. In addition, the algorithm permits intermediate results
of unconverged correction sequences to seed new predictions. This strategy
results in an aggressive optimization of the step length at the cost of
redundancy in the concurrent computation. We present two examples of convoluted
solution curves of high-dimensional systems showing that speed-up by a factor
of two can be attained on a multi-core CPU while a factor of three is
attainable on a small cluster.
