Related papers: Neville's algorithm revisited
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$-…
In this paper we presents an algorithm for finding a solution of the linear nonhomogeneous quaternionic-valued differential equations. Moveover, several examples shows the feasibility of our algorithm.
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
A polynomial algorithm is obtained for the NP-complete linear ordering problem.
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
In this paper we present an estimate of accuracy for a piecewise polynomial approximation of a classical numerical solution to a non linear differential problem. We suppose the numerical solution U is computed using a grid with a small…
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
Given a convergent sequence of nodes we present a one-dimensional-holomorphic-function version of the Newton interpolation method of polynomials. It also generalises the Taylor and the Laurent formula. In other words, we present an…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
We describe an algorithm for the factorization of non-commutative polynomials over a field. The first sketch of this algorithm appeared in an unpublished manuscript (literally hand written notes) by James H. Davenport more than 20 years…
The Numerical Recipes series of books are a useful resource, but all the algorithms they contain cannot be used within open-source projects. In this paper we develop drop-in alternatives to the two algorithms they present for cubic spline…
A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…
A novel algorithm is proposed for the interpolation step of the Guruswami-Sudan list decoding algorithm. The proposed method is based on the binary exponentiation algorithm, and can be considered as an extension of the Lee-O'Sullivan…
We propose a non grid-based interpolation scheme based on the information from the data collected from the vicinity of the query point. As a non-grid-based interpolation, the data points can be distributed randomly in a small region, and…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.