ISSAC

We have proposed error free transformations for floating point numbers[1]-[3]. In the first place, this talk will briely survey this result. Then, the suthor will clarify that this new methodology is very usefull to make efficient error free numerical ...

We propose novel randomized preprocessing techniques for solving linear systems of equations and eigen-solving with extensions to the solution of polynomial and secular equations. According to our formal study and extensive experiments, the approach ...

Cyber-Physical Systems (CPS) are integrations of computation and physical processes. Already now, more or less no new consumer device or industrial machinery does not have some form of integrated computation. Since such systems not only interact with ...

The present talk gives a survey of the DE-Sinc numerical methods (= the Sinc numerical methods, which have been developed by Stenger and his school, incorporated with double-exponential transformations). The DE-Sinc numerical methods have a feature that ...

A Taylor model of a smooth function f over a sufficiently small domain D is a pair (P,I) where P is the Taylor polynomial of f at a point d in D, and I is an interval such that f differs from P by not more than I over D. As such, they represent a hybrid ...

For factoring polynomials in two variables with rational coefficients, an algorithm using transcendental evaluation was presented by Hulst and Lenstra. In their algorithm, transcendence measure was computed. However, a constant c is necessary to compute ...

A bisection method, based on exclusion and inclusion tests, is used to address the nearest univariate gcd problem formulated as a bivariate real minimization problem of a rational fraction.

The paper presents an algorithm, a first implementation and a ...

We present a method to extract factors of multivariate polynomials with complex coefficients in floating point arithmetic. We establish the connection between the reciprocal of a multivariate polynomial and its Taylor expansion. Since the multivariate ...

Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials ...

Recently quantifier elimination (QE) has been of great interest in many fields of science and engineering. In this paper an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm and its variant specially designed for QE are ...

For a proof of the monotone column permanent (MCP)conjecture for dimension 4 it is sufficient to show that 4 polynomials, which come from the permanents of real matrices, are nonnegative for all real values of the variables, where the degrees and the ...

In this paper, we introduce matrix representations of algebraic curves and surfaces for Computer Aided Geometric Design (CAGD). The idea of using matrix representations in CAGD is quite old. The novelty of our contribution is to enable non square ...

Taylor models combine the advantages of numerical methods and algebraic approaches of efficiency, tightly controlled recourses, and the ability to handle very complex problems with the advantages of symbolic approaches, in particularly the ability to be ...

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate ...

By the use of symbolic computation, a problem given by a set of multivariate algebraic relations is often reduced to a univariate algebraic equation which is quite high degree. And, if the roots are required in numbers we generally have to solve the ...

Our unified superfast algorithms for solving Toeplitz, Hankel, Vandermonde, Cauchy, and other structured linear systems of equations with integer coefficients combine Hensel's symbolic lifting and numerical iterative refinement and run in nearly optimal ...

Our main motivation is to analyze and improve factorization algorithms for bivariate polynomials in C[x,y], which proceed by continuation methods.

We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are choosen ...

We present a new algorithm for reconstructing an exact algebraic number from its approximate value by using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on obtaining an ...

We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and inequality ...

In this paper, we present a method for finding zeros of polynomial equations in a given domain. We apply a numerical eigensolver using contour integral for a polynomial eigenvalue problem that is derived from polynomial equations. The Dixon resultant is ...

We present algorithms for multivariate GCD and approximate GCD by modifying Barnett's theorem, which is based on the LU-decomposition of Bézout matrix. Our method is suited for multivariate polynomials with large degrees. Also, we analyze ill-...

Hensel series is an expansion of multivariate algebraic function at a singular point, computed from the defining polynomial by the Hensel construction. The Hensel series is well-structured and tractable, hence it seems to be useful in various ...

We propose three algorithms for approximate factorization of univariate polynomials over Z; the first one uses sums of powers of roots (SPR method), the second one utilizes factor-differentiated polynomials (FD method), and the third one is a robust but ...

Given a univariate polynomial having well-separated clusters of close roots, we give a method of computing close roots in a cluster simultaneously, without computing other roots. We first determine the position and the size of the cluster, as well as ...

In this paper, we study positively invariant sets of a class of nonlinear loops and discuss the relation between these sets and the attractors of the loops. For the canonical Hénon map, a numerical method based on curve fitting is proposed to find a ...

For a certain class of algebraic algorithms, we propose a new method that reduces the number of exact computational steps needed for obtaining exact results. This method is the floating-point interval method using zero rewriting and symbols. Zero ...

In this talk we deal with the problem of parametrizing approximately a perturbed rational affine plane curve implicitly given. We present some of our recent results (see [3], [4], [5], [6]) and we describe our on going research in this context. More ...