We consider the problem of numerically identifying roots of a target function - under the constraint that we can only measure the derivatives of the function at a given point, not the function itself. We describe and characterize two methods for doing this: (1) a local-inversion "inching process", where we use local measurements to repeatedly identify approximately how far we need to move to drop the target function by the initial value over N, an input parameter, and (2) an approximate Newton's method, where we estimate the current function value at a given iteration via estimation of the integral of the function's derivative, using N samples. When applicable, both methods converge algebraically with N, with the power of convergence increasing with the number of derivatives applied in the analysis.

翻译：我们考虑在仅能测量目标函数在给定点的导数（而非函数本身）的约束下，数值识别该函数根的问题。我们描述并刻画了两种实现此目标的方法：（1）局部反演的“渐进逼近过程”——通过局部测量反复确定将目标函数下降初始值的N分之一所需的大致移动距离（其中N为输入参数）；（2）近似牛顿法——利用N个样本估计函数导数的积分，从而估计当前迭代点的函数值。在适用条件下，两种方法均随N呈现代数收敛性，且收敛阶数随分析中所用导数阶数的增加而提高。