A Study on New Muller's Method

When we compute a root of equation F(Jf)=0, Mailer's Method uses three initial approximations XQ, Xi9 and X2 and determines the next approximation Xz by the intersection of the X-axis with the parabola through (Z0, F(Xfi), (Xl} F(Xl))9 and (X2, F(Xt)). The procedure is repeated successively to improve the approximate solution of an equation F(X) =0. Suppose a continuous function Fr defined on the interval [Xo9 Xi~] is given, with F(X$) and F(XJ being opposite signs. In our New-Muller's Method we choose XQ and Xi as the ends of the interval and take another initial approximation X2 as the mid-point of X^ Xi and new approximation X$ is the intersection of the X-axis with a quadratic curve through (Z0,F(Z0)), Qf2,F(Z2)), and (XliF(Xl^. This method is proposed to improve the rate of convergence and calculate faster for reducing the interval. Let us call this method NewMuller's Method in this paper. §1. New-Muiler's Method Let F(X) be continuous function that has a root in an interval Beginning with the initial approximations XQ and X\ under the condition F(XQ) • F(Xi~)<$ an intermediate initial approximation is taken as X2 = ( X0 + XJ /2. Then X3 is the intersection of the X-axis with a quadratic curve G(X)=Q through three point (X0, F(Jf0)), (X2, F(X,)), and (Xl} Next we determine the closed interval / which includes the solution of F(X)=Q, as follow;