非线性方程和一维搜索的反函数解法

用Ｎｅｗｔｏｎ法，无论是解非线性方程，还是进行一维搜索，都只是对函数或导函数进行Ｔａｙｌｏｒ展开取一阶近似．为了提高求解效率欲进行高阶展开则遇到了困难：首先，二阶、三阶展开相应地要解二次、三次代数方程，计算较　麻烦；其次．更高阶展开则不可能求解．本文基于反函数的表达，首次提出了任意阶展开的解法，得到显式表达的解．Ｎｅｗｔｏｎ法成为该方法的一个特例．算例表明反函数解法克服了Ｎｅｗｔｏｎ法有时振荡不收敛的弱点．

Solutions for nonlinear equation and one dimensional search by using inverse functions

The Newton method for solving nonlinear equation or implementing one dimensional search is taking first order approximation of the Taylor's expansion of the function or derivitive function. To raise the efficiency of solutions, higher order's expansions of the function or derivitive function should be taken instead of the first one. But they encountered some difficulties. Formulas of the second or the third order's expansions have to be solved in terms of the second or the third order's algebraic equations whose algorithms are troublesome. Equations of more high order's expansions can not be solved. Based on the inverse function o this paper first proposes solutions of arbitrary order's expansions that are explicit formulas. The Newton method becomes a specific case of methods based on inverse functions. Computational examples show that methods of using inverse functions may overcome the weakness of the Newton method whose algorithm is not stable and not convergent sometimes.