论文标题
反合迭代
Inverse Cubic Iteration
论文作者
论文摘要
对于非线性标量方程，有成千上万的论文。这是一个，要 谈论一种明显的新方法，我称之为“反向立方迭代”（ICI），类似于理查德·布伦特（Richard Brent）的Zeroin方法中的反相反二次迭代。可能的新方法是基于逆函数的切线近似值的立方混合物。我们将这种迭代的数值稳定性重写为两个牛顿步骤的平均值和一个暂时的步骤：每个步骤只需要一个新的功能评估和衍生评估。因此，该方法的总成本仅比牛顿的方法更重要，我们将看到它的订单$ 1+\ sqrt {3} = 2.732 ... $，因此确保为了达到给定的准确性，它通常比牛顿的方法更少，同时使用本质上的相同努力。
There are thousands of papers on rootfinding for nonlinear scalar equations. Here is one more, to talk about an apparently new method, which I call ``Inverse Cubic Iteration'' (ICI) in analogy to the Inverse Quadratic Iteration in Richard Brent's zeroin method. The possibly new method is based on a cubic blend of tangent-line approximations for the inverse function. We rewrite this iteration for numerical stability as an average of two Newton steps and a secant step: only one new function evaluation and derivative evaluation is needed for each step. The total cost of the method is therefore only trivially more than Newton's method, and we will see that it has order $1+\sqrt{3} = 2.732...$, thus ensuring that to achieve a given accuracy it usually takes fewer steps than Newton's method while using essentially the same effort per step.
