基于最小二乘法的方程求根三点公式
A three point formula for finding roots of equations by the method of least squares
AI总结：
提出一种利用最小二乘法拟合三点多项式进行迭代求根的新方法，具有二次收敛速度且无需导数计算。
摘要：
提出了一种新的求根方法，该方法使用涉及三点的数值迭代过程。将需要求根的给定函数 y = f(x) 拟合并近似为形式为 P(x)= a(x-b)^N 的多项式函数，该多项式使用最小二乘法通过三个等距点。使用相同的曲线拟合程序进行连续迭代，以在给定的容差水平内定位根。在每一步迭代中，可以适当改变适用于给定函数形式的曲线幂次 N，以获得更快的收敛速度，并避免可能遇到振荡、发散或越界到无效域的情况。提供了收敛速度的估计。结果表明，该方法具有与牛顿法类似的二次收敛性。提供了示例，展示了该过程以及与割线法和牛顿法的收敛速度比较。该方法不需要计算函数导数。
英文摘要：
A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that passes between three equi-spaced points using the method of least squares. Successive iterations using the same procedure of curve fitting is used to locate the root within a given level of tolerance. The power N of the curve suitable for a given function form can be appropriately varied at each step of the iteration to give a faster rate of convergence and avoid cases where oscillation, divergence or off shooting to an invalid domain may be encountered. An estimate of the rate of convergence is provided. It is shown that the method has a quadratic convergence similar to that of Newton's method. Examples are provided showing the procedure as well as comparison of the rate of convergence with the secant and Newton methods. The method does not require evaluation of function derivatives.
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