Sparse elimination exploits the structure of polynomials by measuring their complexity in terms of Newton polytopes instead of total degree. The sparse, or Newton, resultant generalizes the classical homogeneous resultant and its degree is a function of the mixed volumes of the Newton polytopes. We sketch the sparse resultant constructions of Canny and Emiris and show how they reduce the proble...

Newton’s root finding method applied to a (transcendental) entire function f : C → C is the iteration of a meromorphic function Nf . It is well known that if for some starting value z0, Newton’s method converges to a point ξ ∈ C, then f has a root at ξ. We show that in many cases, if an orbit converges to ξ = ∞ for Newton’s method, then f has a ‘virtual root’ at ∞. More precisely, we show that ...

One approach to computing a square root of a matrix A is to apply Newton's method to the quadratic matrix equation F( X) = X2 A =0. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simpl...

In this paper we consider the application of polynomial root-finding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed r...

In this paper we propose a learning based iterative numerical method for computing a rootξ of a non-linear equation of the form f(x) = 0 in the interval [a,b]. Lag based method through Hermite interpolation modeled root discovery approach is developed and demonstrated in this paper. The new method has been tested for a series of functions considered by several researchers. The numerical experim...

The T -adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T -adic entire function. We give a lower bound for its T -adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T -adic...

The computation of periodic orbits of nonlinear mappings or dynamical systems can be achieved by applying a root-finding method. To determine a periodic solution, an initial guess should be located within a proper area of the mapping or a surface of section of the phase space of the dynamical system. In the case of Newton or Newton-like methods these areas are the basins of convergence correspo...

We present and analyze a new damping approach called backward step control for the globalization of the convergence of Newton-type methods for the numerical solution of nonlinear root-finding problems. We provide and discuss reasonable assumptions that imply convergence of backward step control on the basis of generalized Newton paths in conjunction with a backward analysis argument. In particu...

The problem of obtaining optimal starting values for the calculation of integer roots using Newton's method is considered. It has been shown elsewhere that if relative error is used as the measure of goodness of fit. then optimal results are not obtained when the initial approximation is a best fit. Furthermore, if the so-called logarithmic error instead of the relative error is used in the squ...