Implementing real polyhedral homotopy

  title={Implementing real polyhedral homotopy},
  author={Kisun Lee and Julia Lindberg and Jose Israel Rodriguez},
. We implement a real polyhedral homotopy method using three functions. The first function provides a certificate that our real polyhedral homotopy is applicable to a given system; the second function generates binomial systems for a start system; the third function outputs target solutions from the start system obtained by the second function. This work realizes the theoretical contributions in [6] as easy to use functions, allowing for further investigation into real homotopy algorithms. 



A Polyhedral Homotopy Algorithm For Real Zeros

A homotopy continuation algorithm for finding real zeros of sparse polynomial systems by building on a well-known geometric deformation process, namely Viro's patchworking method, which operates entirely over the real numbers and tracks the optimal number of solution paths.

A polyhedral method for solving sparse polynomial systems

Mixed subdivisions of Newton polytopes are introduced, and they are applied to give a new proof and algorithm for Bernstein's theorem on the expected number of roots, which results in a numerical homotopy with the optimal number of paths to be followed.

HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia

This work presents the Julia package HomotopyContinuation.jl, which provides an algorithmic framework for solving polynomial systems by numerical homotopy continuation by motivating the choice of Julia and how its features allow to improve upon existing software packages with respect to usability, modularity and performance.

HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method

This revision of HOM4PS-2.0 updates its original version in three key aspects: (1) new method for finding mixed cells, (2) combining the polyhedral and linear homotopies in one step, (3) new way of dealing with curve jumping.

The numerical solution of systems of polynomials - arising in engineering and science

Background: Polynomial Systems Homotopy Continuation Projective Spaces Probability One Polynomials of One Variable Other Methods Isolated Solutions: Coefficient-Parameter Homotopy Polynomial

Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation

The structure and design of the software package PHC is described, which features great variety of root-counting methods among its tools and is ensured by the gnu-ada compiler.

Certifying approximate solutions to polynomial systems on Macaulay2

  • Kisun Lee
  • Computer Science, Mathematics
  • 2019
We present the Macaulay2 package NumericalCertification for certifying roots of square polynomial systems. It employs the interval Krawczyk method and α-theory as main methods for certification. The

Complexity and Real Computation

  • L. Blum
  • Mathematics, Computer Science
    Springer New York
  • 1998
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.

Viro's theorem for complete intersections

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Introduction to Interval Analysis

This unique book provides an introduction to a subject whose use has steadily increased over the past 40 years, and provides broad coverage of the subject as well as the historical perspective of one of the originators of modern interval analysis.