Related papers: Solving via modular methods
The decomposition method which makes the parallel solution of the block-tridiagonal matrix systems possible is presented. The performance of the method is analytically estimated based on the number of elementary multiplicative operations…
It has been discovered that linear codes may be described by binomial ideals. This makes it possible to study linear codes by commutative algebra and algebraic geometry methods. In this paper, we give a decoding algorithm for binary linear…
The reduction of a large number of scalar integrals to a small set of master integrals via Laporta's algorithm is common practice in multi-loop calculations. It is also a major bottleneck in terms of running time and memory consumption. It…
We present an alternative method for computing primary decomposition of zero-dimensional ideals over finite fields. Based upon the further decomposition of the invariant subspace of the Frobenius map acting on the quotient algebra in the…
Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.
Specialized function gradient computing hardware could greatly improve the performance of state-of-the-art optimization algorithms, e.g., based on gradient descent or conjugate gradient methods that are at the core of control, machine…
We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation…
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are…
We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a…
The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic…
We present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
Based on recent ideas, stemming from the use of bubbles, we discuss an algorithm for the numerical simulation of the cubic nonlinear Schr{\"o}dinger equation with harmonic potential in any dimension, which could be easily extended to other…
By viewing non-commutative polynomials, that is, elements in free associative algebras, in terms of linear representations, we generalize Horner's rule to the non-commutative (multivariate) setting. We introduce the concept of Horner…
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including…
In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies,…
In this paper, a class of smoothing modulus-based iterative method was presented for solving implicit complementarity problems. The main idea was to transform the implicit complementarity problem into an equivalent implicit fixed-point…
A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths…