Some efficient algorithms for the solution of a single nonlinear equation

High order methods for the numerical solution of nonlinear scalar equations are proposed which are more efficient than known procedures, and a unified approach to various methods suggested in literature is given.

On the simultaneous determination of polynomial roots

On the accurate determination of nonisolated solutions of nonlinear equations

A simple but efficient method to obtain accurate solutions of a system of nonlinear equations with a singular Jacobian at the solution is presented. This is achieved by enlarging the system to a higher dimensional one whose solution in question is isolated. Thus it can be computed e. g. by Newton's method, which is locally at least quadratically convergent and selfcorrecting, so that high accuracy is attainable.

Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations

Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.

On the numerical solution of some finite-dimensional bifurcation problems

We consider numerical methods for solving finite-dimensional bifurcation problems. This paper includes the case of branching from the trivial solution at simple and multiple eigenvalues and perturbed bifurcation at simple eigenvalues. As a numerical example we treat a special rod buckling problem, where the boundary value problem is discretized by the shooting method.

�ber ein Verfahren der Ordnung $$1 + \sqrt 2 $$ zur Nullstellenbestimmung

A new iterative method for solving nonlinear equations is presented which is shown to converge locally withR-order of convergence $$1 + \sqrt 2 $$ at least under suitable differentiability assumptions. The method needs as many function evaluations per step as the classical Newton method.