非线性方程割线法的推广（扩展版）
Generalization of the Secant Method for Nonlinear Equations (extended version)
摘要：
割线法是求解$f(x)=0$形式的非线性方程的一种非常有效的数值方法。它是通过线性插值过程导出的，并且仅采用在逼近$f(x)=0$的根时的$f(x)$的值，因此它每次迭代仅计算$f(x)$一次。在本文中，我们推广了它，用一个合适的$(k+1)$ -点多项式插值代替相关的线性插值，其中$k$是一个至少为2的整数。正如割线法一样，这种推广也具有每次迭代只计算一次$f(x)$的特性。给出了它的误差的封闭形式，并分析了它的收敛阶$s_k$ .我们表明，这种顺序的收敛大于的割线法，它增加对$2$作为$k\to \infty$。（事实上，例如，$s_7=1.9960\cdots$。）对于该方法的效率指数也是如此。我们还证实了理论通过一个说明性的例子。
英文摘要：
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 $k\to \infty$. (Indeed, $s_7=1.9960\cdots$, 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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