Finding zeros of a function is one of the oldest problems in numerical analysis. Newton's method, bisection, secant methods — all are iterative. Start with a guess, evaluate the function, update the guess, repeat. Convergence requires good initial guesses, smooth functions, and enough iterations. The iteration is the cost.
The paper (arXiv:2603.13158, March 2026) extracts exact zero locations from discretized function values on a grid without any iteration. No Newton steps. No convergence criteria. No initial guesses. The zeros are read directly from the data.
The mechanism uses phase information from complex analysis. Even when the function is real-valued, extending it to the complex plane reveals that zeros correspond to points where the phase of the function wraps by 2π around a contour. On a discrete grid, this phase wrapping can be detected by examining the argument (angle) of the function values at grid points. The winding number — how many times the phase wraps — counts the zeros inside each grid cell. The location within the cell is extracted by interpolation of the phase, not of the function values themselves.
The distinction matters: interpolating function values near a zero is poorly conditioned because the function is small and the relative error is large. Interpolating the phase near a zero is well-conditioned because the phase varies smoothly and the winding is a topological invariant — it doesn't depend on the magnitude of the function values, only on their angles.
The structural lesson: the same information (function values on a grid) supports two different representations (magnitude and phase), and the representation that seems less natural for a real-valued problem (phase, which requires extending to the complex plane) is the one that solves the problem without iteration. The iteration in traditional methods compensates for working in the wrong representation. Switch representations, and the iteration becomes unnecessary.