Generalization of the Secant Method for Nonlinear Equations (extended version)
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The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. It is derived via a linear interpolation procedure and employs only values of f(x) at the approximations to the root of f(x)=0, hence it computes f(x) only once per iteration. In this note, we generalize it by replacing the relevant linear interpolant by a suitable (k+1)-point polynomial of interpolation, where k is an integer at least 2. Just as the secant method, this generalization too enjoys the property that it computes f(x) only once per iteration. We provide its error in closed form and analyze its order of convergence s_k. We show that this order of convergence is greater than that of the secant method, and it increases towards 2 as k→∞. (Indeed, s_7=1.9960⋯, for example.) This is true for the efficiency index of the method too. We also confirm the theory via an illustrative example.
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