# Root finding via local measurement

## TL;DR

This paper introduces two novel numerical methods for root finding that rely solely on derivative measurements at a point, avoiding direct function evaluations, and analyzes their convergence properties.

## Contribution

It proposes and characterizes two new derivative-only root-finding algorithms, expanding the toolkit for scenarios with limited function access.

## Key findings

- Both methods converge algebraically with respect to N.
- Convergence rate improves with the number of derivatives used.
- Methods are applicable under certain conditions for derivative measurements.

## Abstract

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.