Related papers: Neville's algorithm revisited
An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving…
Equivalence between algebraic equations of motion may be detected by using a $p$-adic method, methods using factorization and linear algebra, or by systematic computer search of suitable Tschirnhausen transformations. Here, we show standard…
The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form.
Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of…
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
The over-parameterized models attract much attention in the era of data science and deep learning. It is empirically observed that although these models, e.g. deep neural networks, over-fit the training data, they can still achieve small…
In this paper, we introduce the neural empirical interpolation method (NEIM), a neural network-based alternative to the discrete empirical interpolation method for reducing the time complexity of computing the nonlinear term in a reduced…
We consider the problem of recovering (that is, interpolating) and identity testing of a "hidden" monic polynomial $f$, given an oracle access to $f(x)^e$ for $x\in{\mathbb F_q}$ (extension fields access is not permitted). The naive…
In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the…
This paper provides error analyses of the algorithms most commonly used for the evaluation of the Chebyshev polynomial of the first kind $T_N(x)$. Some of these algorithms are shown to be backward stable. This means that the computed value…
In this paper we present an efficient algorithm for bivariate interpolation, which is based on the use of the partition of unity method for constructing a global interpolant. It is obtained by combining local radial basis function…
We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari for interpolating polynomials over rings with characteristic zero,…
Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is…
Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a…
Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement…
Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper…
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced…
Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…
The multigrid algorithm is an efficient numerical method for solving a variety of elliptic partial differential equations (PDEs). The method damps errors at progressively finer grid scales, resulting in faster convergence compared to…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…