An acceleration procedure of Regula Falsi method.

Resumen

The regula falsi method is one of the oldest methods for the solution of f(x)=0; its important advantages have been realized for use on computers. But it has only linear convergence. The authors regula falsi, xn=xn−1−f(xn−1)(λ−xn−1)/(f(λ)−f(xn−1)), is generalized in this paper by replacing f(λ) with a parameter μ: xn=xn−1−f(xn−1)(λ−xn−1)/(μ−f(xn−1)). Then they apply a geometric procedure to determine μn−1 for the acceleration, and obtain the acceleration {yn} from {xn} by yn=xn−1−f(xn−1)(λ−xn−1)/(μn−1−f(xn−1)). From this they derive a new acceleration for {xn}: yn=xn−1−f(xn−1)μ/(f′(xn−1)(μ−f(xn−1))), and prove the sequence {yn} has, at least, quadratic convergence. Finally, from it they define a new iterative process: tn,μ=tn−1,μ−f(tn−1,μ)μ/(f′(tn−1,μ)(μ−f(tn−1,μ))). To give conditions for the convergence of {tn,μ} they consider the degree of logarithmic convexity of functions. All the results improve on the Newton method, but the conditions for convergence applied to f(x) are more complicated than in the Newton method.