Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complementary dimension. We represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this talk we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better stepsize control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.
This is joint work with Yun Guan.
FoCM 2011 -- Real Number Complexity Workshop, Budapest, Hungary, 4-6 July 2011.