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It is well known that, using fast algorithms for polynomial multiplication and division, evaluation of a polynomial $F \in \mathbb{C}[x]$ of degree $n$ at $n$ complex-valued points can be done with $\tilde{O}(n)$ exact field operations in…
In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain…
This paper belongs to a group of work in the intersection of symbolic computation and group analysis aiming for the symbolic analysis of differential equations. The goal is to extract important properties without finding the explicit…
Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of…
We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is…
A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial…
The paper proposes, an algorithm to produce novel m-point (for any integer m>=2) binary non-stationary subdivision scheme. It has been developed using uniform trigonometric B-spline basis functions and smoothness is being analyzed using the…
Manipulating Hamiltonians governing physical systems has found a broad range of applications, from quantum chemistry to semiconductor design. In this work, we provide a new way of manipulating Hamiltonians, by transforming their eigenvalues…
In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline the general methodology and then apply it to five distinct experiments.…
In this work we present and study an iterative algorithm used to asymptotically solve nonlinear differential equations. This algorithm (Iterative First Order HAM or IFOHAM) is based on the first order equation of the Homotopy Analysis…
This paper considers the problem of computing the operator norm of a linear map between finite dimensional Hilbert spaces when only evaluations of the linear map are available and under restrictive storage assumptions. We propose a…
This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only…
Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such…
Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for…
A new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented in Mathematica. The program condens.m automatically carries out the…
We investigate the problem of recovering coefficients in scalar nonlinear ordinary differential equations that can be exactly linearized. This contribution builds upon prior work by Lyakhov, Gerdt, and Michels, which focused on obtaining a…
We analyze a new algorithm for probability forecasting of binary observations on the basis of the available data, without making any assumptions about the way the observations are generated. The algorithm is shown to be well calibrated and…
Thom polynomial describes the cohomology class Poincar\'e dual to the locus of particular singularity of a generic holomorphic map. In this paper we derive a closed formula for the generating function of its coefficients. The method is…
The principal minors of a tridiagonal matrix satisfy two-term and three-term recurrences [1, 2]. Based on these facts, the current article presents a new efficient and reliable hybrid numerical algorithm for evaluating general n-th order…
Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an…