A method of order four for finding multiple zeros of nonlinear functions is developed. The method is based on Jarratt’s 4 fifth-order method (for simple roots) and it requires one evaluation of the function and three evaluations of the derivative. The 5 informational efficiency of the method is the same as previously developed schemes of lower order. For the special case of double 6 root, we fo...

We present a combination of two novel algorithms that accurately calculate multiple roots of general polynomials. For a given multiplicity structure and initial root estimates, Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm I, we ...

Newton’s iteration has quadratic convergence for simple roots. We present a Newton-based iteration scheme with quadratic convergence for multiple roots of systems of analytic functions. This is a report on work in progress. 1. Newton Iteration, Approximate Roots and γ-Theorems 1.1. Newton Iteration. Let f : Cn → Cn be an analytic function. Newton’s method for solving f = 0 consists in approxima...

In this paper, we present a new fourth-order modification of Newton’s method for multiple roots, which is based on existing second -order. Modification of Newton’s multiple root-finding methods. Some numerical examples illustrate that the new method is more efficient and performs better than other methods for multiple roots.

In this paper we present a new method to find simple or multiple roots of functions in a finite interval. In this method using bisection method we can find an interval such that this function is one to one on it, thus we can transform problem of finding roots in this interval into an ordinary differential equation with boundary conditions. By solving this equation using collocation method we ca...

In this paper, we present two new families of iterative methods for multiple roots of nonlinear equations. One of the families require one-function and two-derivative evaluation per step, and the other family requires two-function and one-derivative evaluation. It is shown that both are third-order convergent for multiple roots. Numerical examples suggest that each family member can be competit...

Algebraic eigenvalues with associated left and right eigenvectors (nearly) orthogonal, and polynomial roots that are multiple, have been known to be sensitive to perturbations in numerical computation and thereby ill-conditioned. By constructing an extended polynomial system, it is proved that, under certain conditions, traditionally deened ill-conditioned eigenvalues and polynomial roots can b...