Related papers: On the approximate roots of polynomials
In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.
Univariate polynomial root-finding is both classical and 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 polynomial…
We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are…
Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph…
Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some…
In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…
Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the…
We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d. coefficients in $\mathbb{Z}_p$, we obtain an estimate for the expected number of roots of this polynomial. In…
In this paper, we propose a method to obtain a constrained approximation of a rational B\'{e}zier curve by a polynomial B\'{e}zier curve. This problem is reformulated as an approximation problem between two polynomial B\'{e}zier curves…
Consider a primitive polynomial $f$ in two variables, thought of as a map from the affine plane to the affine line. We study the minimimal compactification of $f$; from our result one deduces in particular that if one of the fibers of $f$…
We construct parametric families of (monic) reducible polynomials having two roots very close to each other.
A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that…
We introduce an asymmetric operator of generalised translation, define the generalised modulus of smoothness by its means, and obtain the direct and inverse theorems in approximation theory for it.
The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes…
We consider the problem of numerically identifying roots of a target function - under the constraint that we can only measure the derivatives of the function at a given point, not the function itself. We describe and characterize two…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the…
In this paper we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. The results include approximation…
We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f in K[x,y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a…
Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum…