The database of polynomial systems

The demonstration database is an essential part of PHCpack. Motivations are plentiful:

The database is tailored towards PHCpack. Besides the peculiar format of solution lists, this implies that the systems often have floating-point input coefficients and the focus is on approximating the isolated roots. Moreover, one criterion for entering the database is that it can be solved with PHCpack.

We tried to find original references, sometimes a piece of Maple code accompanies the derivation of the polynomial equations. An exciting part of the field of polynomial system solving is that one encounters applications as diverse as game theory, mechanical design, chemical reactions, control of linear systems, etc... We hope that the relevance of these applications will help to elevate our field to approximate the status currently enjoyed by areas such as Linear Algebra, Optimization, or Differential Equations.

We warn that a general-purpose solver will in general not be competitive against methods tailored for a specific application. Homotopies unable to exploit structure usually fail and end up squandering computer resources.

Some new test examples...

Some examples to illustrate Newton's method with deflation for isolated singularities.

The files in the left column below contain the original formulation of the polynomial system, along with a list of initial approximations for the isolated solutions. These approximations have been found at the end of solution paths defined by a homotopy. The other columns below contain the deflated systems along with their solutions. The number following the "_d" in the name is the deflation step while the number following the "_R" in the name is the numerical rank of the Jacobian matrix at the isolated root before the deflation. Multiple deflations may be needed to characterize a multiple root.

Some examples to illustrate the monodromy breakup algorithm to decompose algebraic sets into irreducible components.

This material is based upon work supported by the National Science Foundation under Grants No. 9804846, 0105739, and 0134611. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.