Related papers: Decomposable sparse polynomial systems
This paper applies the Thomas decomposition technique to nonlinear control systems, in particular to the study of the dependence of the system behavior on parameters. Thomas' algorithm is a symbolic method which splits a given system of…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices…
This note introduces the Macaulay2 package SchurVeronese, which gathers together data about Veronese syzygies and makes it readily accessible in Macaulay2. In addition to standard Betti tables, the package includes information about the…
In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
The Macaulay2 package RealRoots provides symbolic methods to study real solutions to systems of polynomial equations. It updates and expands an earlier package developed by Grayson and Sottile in 1999. We provide mathematical background and…
We present an adaptation of the so-called structural method \cite{CMM23} for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem…
We discuss the parallelization of algorithms for solving polynomial systems symbolically by way of triangular decomposition. Algorithms for solving polynomial systems combine low-level routines for performing arithmetic operations on…
This introduces Rees algebras and some of their uses with illustrations via version 2.0 of the Macaulay2 package ReesAlgebra.m2.
We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result…
We present a highly scalable algorithm for multiplying sparse multivariate polynomials represented in a distributed format. This algo- rithm targets not only the shared memory multicore computers, but also computers clusters or specialized…
Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic…
The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is explored. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy.…
We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly to identify rational and geometric points in a variety over a finite field. We provide tools to estimate the dimension of a variety. We also present methods to…
In this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study…
The polynomial algebra is a deformed SU(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Meanwhile, we find a series of physics systems corresponding with polynomial algebra with different…
In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by…
Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques…
This work proposes a research problem of finding sparse solution of undetermined Linear system with some applications. Two approaches how to solve the compressive sensing problem: using l_1 approach , the l_q approach with 0 < q < 1.…