Related papers: Accurate Evaluation of Polynomials
The study addresses the problem of precision in floating-point (FP) computations. A method for estimating the errors which affect intermediate and final results is proposed and a summary of many software simulations is discussed. The basic…
One of the main open problems in the qualitative theory of real planar differential systems is the study of limit cycles. In this article, we present an algorithmic approach for detecting how many limit cycles can bifurcate from the…
In this paper we present a new algorithm for Polynomial Linear System Solving (via evaluation/interpolation) with errors. In this scenario, errors can occur in the black box evaluation step. We improve the bound on the number of errors that…
This work proposes a new method for computing acceptance regions of exact multinomial tests. From this an algorithm is derived, which finds exact p-values for tests of simple multinomial hypotheses. Using concepts from discrete convex…
We develop a simple two-step algorithm for enclosing Chebyshev expansions whose cost is linear in terms of the polynomial degree. The algorithm first transforms the expansion from Chebyshev to the Laurent basis and then applies the interval…
In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
Using elementary methods, we define and derive a particular weighted average of the trapezoidal and composite trapezoidal rules and show that this approximation, as well as its composite, is straightforward in computation. This…
Polynomial regression is widely used and can help to express nonlinear patterns. However, considering very high polynomial orders may lead to overfitting and poor extrapolation ability for unseen data. The paper presents a method for…
In computer aided geometric design a polynomial is usually represented in Bernstein form. This paper presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The…
An iterative formula based on Newton Method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method…
We present double pooling, a simple, easy-to-implement variation on test pooling, that in certain ranges for the a priori probability of a positive test, is significantly more efficient than the standard single pooling approach (the Dorfman…
Newton's method for polynomial root finding is one of mathematics' most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the…
We first review existing sequential methods for estimating a binomial proportion. Afterward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and confidence…
We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions…