Quasi-analytical root-finding for non-polynomial functions

Root-finding and Root-refining for a Polynomial Equation

Polynomial root-finders usually consist of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iteration. The efficiency of computing an initial approximation resists formal study, and the users rely on empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q whe...

Polynomial Root-Finding Algorithms and Branched Covers

Introduction. The problem of devising optimal methods for numerically approximating the roots of a polynomial has been of interest for several centuries, and is far from solved. There are numerous recent works on root-finding algorithms and their cost, for example, the work of Jenkins and Traub [JT70], Renegar [Ren87], Schönhage [Sch82], and Shub and Smale [SS85, SS86, Sma85]. This list is far ...

A Basic Family of Iteration Functions for Polynomial Root Finding and Its Characterizations

Let p(x) be a polynomial of degree n 2 with coe cients in a sub eld K of the complex numbers. For each natural number m 2, let L m (x) be the m m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 1, its j-th subdiagonal entries are p (j) (x)=j!. For i = 1; 2, let L (i) m (x) be the matrix obtained from L m (x) by deleting its rst i rows and its last i columns...

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

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...

Polynomial Hermite Interpolation for Root Finding and Event Location