Related papers: Rigorous Multiple-Precision Evaluation of D-Finite…
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…
In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types…
In the last decade major steps towards an algorithmic treatment of orthogonal polynomials and special functions (OP & SF) have been made, notably Zeilberger's brilliant extension of Gosper's algorithm on algorithmic definite hypergeometric…
We study generalized log-sine integrals at special values. At $\pi$ and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at $\pm1$. For general arguments we present algorithmic evaluations involving…
In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs…
We present an adaptive algorithm for effectively solving rough differential equations (RDEs) using the log-ODE method. The algorithm is based on an error representation formula that accurately describes the contribution of local errors to…
We consider algebraic ordinary differential equations (AODEs) and study their polynomial and rational solutions. A sufficient condition for an AODE to have a degree bound for its polynomial solutions is presented. An AODE satisfying this…
This paper depicts an algorithm for solving the Decision Boolean Satisfiability Problem using the binary numerical properties of a Special Decision Satisfiability Problem, parallel execution, object oriented, and short termination. The two…
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
The SageManifolds project aims at extending the mathematics software system Sage towards differential geometry and tensor calculus. Like Sage, SageManifolds is free, open-source and is based on the Python programming language. We discuss…
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original…
The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
A new expression for solving homogeneous linear ODEs based on a generalization of the Volterra composition was recently introduced. In this work, we extend such an expression, showing that it corresponds to inverting an infinite matrix.…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
We present an algorithm which computes the $D^{th}$ term of a sequence satisfying a linear recurrence relation of order $d$ over a field $K$ in $O( \mathsf{M}(\bar d)\log(D) + \mathsf{M}(d)\log(d))$ operations in $K$, where $\bar d \leq d$…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
We revisit the classical problem of construction of a fundamental system of solutions to a linear ODE whose elements remain analytic and linearly independent for all values of the roots of the characteristic polynomial.