Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. We have experimented with criteria to monitor the Jacobian in order not to miss any singular solutions along a path. In case of higher order singularities more accurate predictors are needed, otherwise we do not get in the range for which reconditioning methods such as deflation can be applied. Our methods are implemented in the software package PHCpack and applied to a wide range of polynomial systems arising in various fields of science and engineering.
Joint work with Kathy Piret.
ACA 2009 Session on Algorithms for Parametric Systems and their Applications. ETS, Montreal, Canada, 25-28 June 2009.