Finding All Isolated Solutions To Polynomial Systems Using Hompack

Although the theory of polynomial continuation has been established for over a decade (following the work of Garcia, Zangwill, and Drexler), it is difficult to solve polynomial systems using continuation in practice. Divergent paths (solutions at infinity), singular solutions, and extreme scaling of coefficients can create catastrophic numerical problems. Further, the large number of paths that typically arise can be discouraging. In this paper we summarize polynomial-solving homotopy continuation and report on the performance of three standard path-tracking algorithms (as implemented in HOMPACK) in solving three physical problems of varying degrees of difficulty. Our purpose is to provide useful information on solving polynomial systems; including specific guidelines for homotopy construction and parameter settings. The m-homogeneous strategy for constructing polynomial homotopies is outlined, along with more tradition approaches. Computational comparisons are included to illustrate and contrast the major HOMPACK options. The conclusions summarize our numerical experience and discuss areas for future research

A Special Homotopy Continuation Method For A Class of Polynomial Systems

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety

Homotopies for solving polynomial systems within a bounded domain

AbstractThe problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouche's theorem by homotopy continuation concepts yields a new class of homotopy methods, the so-called regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. As the dominant part has a sparser structure, it is easier to solve. It will be used as start system in the regional homotopy. The paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain. Applications illustrate the usefulness of the approach

Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.Comment: 36 page

Numerical polynomial homotopy continuation method to locate all the power flow solutions

The manuscript addresses the problem of finding all solutions of power flow equations or other similar non-linear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly the direct methods for transient stability analysis and voltage stability assessment. Here, the authors introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. Since finding real solutions is much more challenging, first the authors embed the real form of power flow equation in complex space, and then track the generally unphysical solutions with complex values of real and imaginary parts of the voltages. The solutions converge to physical real form in the end of the homotopy. The so-called gamma-trick mathematically rigorously ensures that all the paths are well-behaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelisable. The authors demonstrate the technique performance by solving several test cases up to the 14 buses. Finally, they discuss possible strategies for scaling the method to large size systems, and propose several applications for security assessments