求解方程的一类迭代公式

A Class of Iterative Formulae for Solving Equations

中文关键词:  非线性方程  迭代函数  收敛阶数  Newton方法  Halley方法.

英文关键词:Non-linear equation  iterative function  order of convergence  Newton's method  Halley's method.

中文摘要:

      基于Newton迭代函数、求重根的Newton迭代函数和Halley迭代函数, 本文提出了求解单变量方程的一类迭代公式，并且证明了这些迭代公式至少是二阶收敛的.最后,通过数值例子比较了Newton方法、Halley方法以及本文提出的方法.数值结果表明如果选择合适的参数函数~$\lambda (x)$ 和 $\mu (x)$,能使得本文的方法收敛,从而显示了该方法的可行性和有效性.

英文摘要:

      Using the forms of Newton iterative function, the iterative function of Newton's method to handle the problem of multiple roots and the Halley iterative function, we give a class of iterative formulae for solving equations in one variable in this paper and show that their convergence order is at least quadratic. At last we employ our methods to solve some non-linear equations and compare them with Newton's method and Halley's method. Numerical results show that our iteration schemes are convergent if we choose two suitable parametric functions $\lambda (x)$ and $\mu (x)$. Therefore, our iteration schemes are feasible and effective.