Related papers: Obtaining Exact Interpolation Multivariate Polynom…
Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term numerical emphasizes that a numerical solution is computed. The method consists in replacing the right hand…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here…
In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate…
In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great…
We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad…
Sampling from multimodal distributions is a challenging task in scientific computing. When a distribution has an exact symmetry between the modes, direct jumps among them can accelerate the samplings significantly. However, the…
We propose a simple interpolation-based method for the efficient approximation of gradients in neural ODE models. We compare it with the reverse dynamic method (known in the literature as "adjoint method") to train neural ODEs on…
The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Pade and other interpolation schemes. A unified view of these various…
The roots of any polynomial of degree m with complex integer coefficients can be computed by manipulation of sequences made from distinct symbols and counting the different symbols in the sequences. This method requires only primitive…
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…
Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded…
We describe an algorithm for fast multiplication of skew polynomials. It is based on fast modular multiplication of such skew polynomials, for which we give an algorithm relying on evaluation and interpolation on normal bases. Our…
Motivated by a recently introduced HIMMO key distribution scheme, we consider a modification of the noisy polynomial interpolation problem of recovering an unknown polynomial $f(X) \in Z[X]$ from approximate values of the residues of $f(t)$…
In this paper we investigate polynomial interpolation using orthogonal polynomials. We use weight functions associated with orthogonal polynomials to define a weighted form of Lagrange interpolation. We introduce an upper bound of error…
We present two new methods for multivariate exponential analysis. In [7], we developed a new algorithm for reconstruction of univariate exponential sums by exploiting the rational structure of their Fourier coefficients and reconstructing…
This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a…
Nonlinear interpolants have been shown useful for the verification of programs and hybrid systems in contexts of theorem proving, model checking, abstract interpretation, etc. The underlying synthesis problem, however, is challenging and…