We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an algorithm to approximate the function by a polynomial without using higher order differentiability information, which depends essentially on integrability. Moreover, we extend the method to a system of equations if the Jacobian determinant does not vanish. This is a robust method for implicit functions that are not differentiable to higher order. Additionally, we present two numerical experiments to verify the theoretical results.

中文翻译：

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一种不使用高阶可微信息的多项式逼近隐函数的算法

我们考虑一个多变量方程，其中偏导数在一点处不会消失。隐函数定理提供了方程函数的局部存在性和唯一性。在本文中，我们提出了一种通过多项式逼近函数的算法，而不使用高阶可微性信息，这本质上取决于可积性。此外，如果雅可比行列式不消失，我们将该方法扩展到方程组。对于不可微分到高阶的隐式函数，这是一种稳健的方法。此外，我们提出了两个数值实验来验证理论结果。