Related papers: A Blackbox Polynomial System Solver on Parallel Sh…
The Macaulay2 package PHCpack.m2 provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The…
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…
We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial $f \in K[x_1 , . . . , x_n]$ (where $K \subseteq \mathbb{C}$) of…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which…
Many parallel algorithms use at least linear auxiliary space in the size of the input to enable computations to be done independently without conflicts. Unfortunately, this extra space can be prohibitive for memory-limited machines,…
An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain…
We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…
The Sequential Multiple Knapsack Problem is a special case of Multiple knapsack problem in which the items sizes are divisible. A characterization of the optimal solutions of the problem and a description of the convex hull of all the…
A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular…
A general class of polynomial remainder codes is considered. Such codes are very flexible in rate and length and include Reed-Solomon codes as a special case. As an extension of previous work, two joint error-and-erasure decoding approaches…
We introduce an algorithm to decompose orthogonal matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
We introduce a backward stable algorithm for computing the CS decomposition of a partitioned $2n \times n$ matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm computes two $n \times n$ polar decompositions…
In view of the existing limitations of sequential computing, parallelization has emerged as an alternative in order to improve the speedup of numerical simulations. In the framework of evolutionary problems, space-time parallel methods…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…