Why Polyhedra Matter in Non-Linear Equation Solving
نویسندگان
چکیده
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and assume no background in algebraic geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein’s Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area — a quantity intimately related to counting complex roots in the plane. (3) Illustration of the connection between polyhedral methods, amoeba theory, toric varieties, and discriminants. We thus cover most of the theory preceding polyhedral homotopy and toric (a.k.a. sparse) resultant-based methods for solving systems of multivariate polynomial equations
منابع مشابه
Modified homotopy perturbation method for solving non-linear oscillator's equations
In this paper a new form of the homptopy perturbation method is used for solving oscillator differential equation, which yields the Maclaurin series of the exact solution. Nonlinear vibration problems and differential equation oscillations have crucial importance in all areas of science and engineering. These equations equip a significant mathematical model for dynamical systems. The accuracy o...
متن کاملComparison of The LBM With the Modified Local Crank-Nicolson Method Solution of Transient Two-Dimensional Non-Linear Burgers Equation
Burgers equation is a simplified form of the Navier-Stokes equation that represents the non-linear features of it. In this paper, the transient two-dimensional non-linear Burgers equation is solved using the Lattice Boltzmann Method (LBM). The results are compared with the Modified Local Crank-Nicolson method (MLCN) and exact solutions. The LBM has been emerged as a new numerical method for sol...
متن کاملPreconditioned Generalized Minimal Residual Method for Solving Fractional Advection-Diffusion Equation
Introduction Fractional differential equations (FDEs) have attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or numerical scheme may be a good approach, particularly, the schemes in numerical linear algebra for solving ...
متن کاملA programming method to estimate proximate parameters of coal beds from well-logging data using a sequential solving of linear equation systems
This paper presents an innovative solution for estimating the proximate parameters of coal beds from the well-logs. To implement the solution, the C# programming language was used. The data from four exploratory boreholes was used in a case study to express the method and determine its accuracy. Then two boreholes were selected as the reference, namely the boreholes with available well-logging ...
متن کامل