A new class of quite elementary closed-form integral formulae for roots of nonlinear equations

摘要

A new class of analytical formulae (containing one integral term) for the determination of simple roots of nonlinear algebraic or transcendental equations inside finite intervals is proposed. All of these formulae are very elementary, and they are based on the definition of the sign function (or, equivalently, the absolute-value function) appearing in them. Moreover, each one of these formulae results from the previous one by the method of integration by parts, the first one being derived simply by inspection. An application to the classical Kepler's equation is made, and numerical results, verifying the validity of these formulae and showing their numerical effectiveness, are presented. The advantages of the method are reported, and its rate of convergence is established (when numerical integration rules are used).