On some numerical integration methods for solving algebraic equations

ThIs paper examines theiterative arise from the numerical methods for the integration approach [3,5,7] to nonlinear algebraic equations. Convergence of some multistep methods and single-step methods has been demonstrated in [1,6]. While these methods do not require as good an initial approximation as Newton-Raphson method, they have not been widely implemented due to (a) unavailability of convergence rate, (b) a lack of rapid convergence close to the solution. In this paper we shall show that the problems of (a) and (b) are closely related to the stepsize for numerical integration. Specifically, rapid convergence can be attained with a judicious choice of stepsize and the resulting rate of convergence is either quadratic or weaker depending on the numerical method. It follows that of Euler's method, third order Runge-Kutta method and the two-correction trapezoidal rule are quadratically convergent with stepsizes 1, 1.596071638 and 1.295597743 approximately. The first result is obvious since the Euler's method with stepsize 1 reduces to the Newton-Raphson method. The findings on non-Euler methods provide significant insight into the numerical integration approach to algebraic problems.Technical report DCS-TR-2