Related papers: Accurate Evaluation of Polynomials
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…
Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one…
We generalize univariate multipoint evaluation of polynomials of degree n at sublinear amortized cost per point. More precisely, it is shown how to evaluate a bivariate polynomial p of maximum degree less than n, specified by its n^2…
The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes…
The paper deals with the analytic complexity of solutions to bivariate holonomic hypergeometric systems of the Horn type. We obtain estimates on the analytic complexity of Puiseux polynomial solutions to the hypergeometric systems defined…
In numeric-intensive computations, it is well known that the execution of floating-point programs is imprecise as floating-point arithmetic incurs round-off errors. Although round-off errors are small for a single floating-point operation,…
The performance of numerical micromagnetic models is limited by the demagnetizing field computation, which typically accounts for the majority of the computation time. For magnetization dynamics simulations explicit evaluation methods are…
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 give several new algorithms for dense polynomial multiplication based on the Kronecker substitution method. For moderately sized input polynomials, the new algorithms improve on the performance of the standard Kronecker substitution by a…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum…
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…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is…
In this paper, we study two ways of evaluating iterated Ore polynomials. We provide many examples and compare these evaluations. We use the evaluation maps to construct Reed-Muller codes and compute explicitly some of the data that are…
The numerical construction of polynomials in the product representation (as used for instance in variants of the multiboson technique) can become problematic if rounding errors induce an imprecise or even unstable evaluation of the…
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…
We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…