Next: Polynomial Continuation Up: Reference Manual Previous: Root Counts and Start

The artificial-parameter homotopy

The artificial-parameter homotopy has the following form:

with a start system, and the target system.

The homotopy parameter k
determines the power of the continuation parameter t. Taking k>1 has as effect that at the beginning and at the end of the continuation, changes in t do not change the homotopy as much as with a homotopy that is linear in t so that paths are better to follow. The default value k=2 is recommended.
The homotopy parameter a
ensures the accessibility and regularity of the solution paths, i.e.: by choosing a random complex number for a, all paths are regular and do not diverge for t<1.

The target value
for the continuation parameter t is by default 1. To create stepping stones in the continuation stage, it is possible to let the continuation stop at t<1, for instance at t = 0.9 or even at a complex value for t. The solutions at t<1 will serve at start solutions to take up the continuation in a later stage. In this stage, the same homotopy parameters k and a must be used.

A projective transformation
of the homotopy and start solutions makes the equations in the polynomials homogeneous and adds a random hyperplane. The vectors of the start solutions are extended with an additional unknown. For solutions at infinity, this additional unknown equals zero.

Next: Polynomial Continuation Up: Reference Manual Previous: Root Counts and Start
Jan Verschelde
3/7/1999