论文标题
非线性方程的割线方法的概括（扩展版本）
Generalization of the Secant Method for Nonlinear Equations (extended version)
论文作者
论文摘要
SECANT方法是一个非常有效的数值过程，用于求解$ f（x）= 0 $的非线性方程。它是通过线性插值过程得出的，并且仅在$ f（x）= 0 $的根上使用$ f（x）$的值，因此它计算$ f（x）$每次迭代一次。在本说明中，我们通过将相关线性插值替换为合适的$（k+1）$ - 插值点多项式来概括它，其中$ k $是一个整数至少2个。就像SCANT方法一样，这种概括也很喜欢它仅计算$ f（x）$的属性。我们以封闭形式提供其错误，并分析其收敛顺序$ s_k $。我们表明，这种融合顺序大于割线方法的顺序，并且增加到$ 2 $为$ k \ to \ infty $。 （实际上，$ s_7 = 1.9960 \ cdots $。）对于该方法的效率索引也是如此。我们还通过一个说明性示例确认了理论。
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.
