On Hessian limit directions along non-oscillating gradient trajectories

TL;DR

This paper investigates the behavior of Hessian matrices along non-oscillating gradient trajectories of real analytic functions, revealing eigenvector limits and providing estimates, with implications for understanding function dynamics at critical points and infinity.

Contribution

It establishes that the limit of secants along such trajectories aligns with Hessian eigenvectors, extending results to infinity for globally subanalytic functions.

Findings

Limit secants are eigenvectors of the Hessian limit.

Results apply both at finite critical points and at infinity.

Provides estimates along the gradient trajectories.

Abstract

Given a non-oscillating gradient trajectory G of a real analytic function f, we show that the limit v of the secants at the limit point O of G along the trajectory G is an eigen-vector of the limit of the direction of the Hessian matrix Hess (f) at O along G. The same holds true at infinity if the function is globally subanalytic. We also deduce some interesting estimates along the trajectory.