Adaptive step-size control

 

predictor type: 

We distinguish between predictors for the solution vector $\bf x$and the continuation parameter t. The secant predictor  for $\bf x$ is based on linear extrapolation, whereas the tangent predictor  uses the Jacobian matrix to extrapolate. For t, three different predictors are provided. The real predictor  simply augments t with the step size. The complex predictor  takes values in the complex plane to circumvent values of t for which the corresponding system is badly conditioned. The geometric predictor  generates values for t so that the distances to the target value behave like a geometric series. The latter predictor is useful in a polyhedral end game.

minimum step size:  

This is a lower bound on the step size for the continuation parameter. Continuation for the path being followed stops when a required reduction of the step size yields a step length that is lower than the minimum step size.

maximum step size: 

This is an upper bound on the step size for the continuation parameter. An expansion of the step size that yields a step length that is larger than this threshold sets the step size to this threshold.

reduction factor for step size: 

When the corrector does not manage to reach the required accuracy within the allowed number of iterations, the step size will be reduced by multiplication with the reduction factor.

expansion factor for step size: 

When the corrector has reached the required accuracy within the allowed number of iterations, for a number of consecutive times higher than the expansion threshold, the step size will be enlarged by multiplication by the expansion factor.

expansion threshold: 

Before enlarging the step size along a path, the number of consecutive successful steps must pass the expansion threshold. A step is successful when the corrector manages to reach the required accuracy within the allowed number of iterations. At the end of the path, this threshold bounds the number of consecutive equal guesses of the winding number m before the polyhedral end game can modify it.