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On Hessian Limit Directions along Gradient Trajectories
Published online by Cambridge University Press: 20 November 2018
Abstract
Given a non-oscillating gradient trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ of a real analytic function
$f$, we show that the limit
$v$ of the secants at the limit point
$0$ of
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ along the trajectory
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ is an eigenvector of the limit of the direction of the Hessian matrix Hess
$\left( f \right)$ at
$0$ along
$\left| \text{ }\!\!\gamma\!\!\text{ } \right|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.
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- Research Article
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- Copyright © Canadian Mathematical Society 2013