Inflection point

Summary

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class C2 (f, its first derivative f, and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vice versa as f'' is continuous; an inflection point of the curve is where f'' = 0 and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called

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https://en.wikipedia.org/wiki/Inflection_point

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Étudier les concepts fondamentaux d'analyse, et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

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We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit points are not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this explicitly for an existing optimization algorithm on bounded-rank matrices. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond bounded-rank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.

SPRINGER HEIDELBERG2022

Low rank differential equations for Hamiltonian matrix nearness problems

Daniel Kressner

For a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems. They are closely related to the problem of determining extremal points of Hamiltonian pseudospectra. We obtain a characterization of optimal perturbations, which turn out to be of low rank and are attractive stationary points of low-rank differential equations that we derive. We use a two-level approach, where in the inner level we determine extremal points of the Hamiltonian -pseudospectrum for a given by following the low-rank differential equations into a stationary point, and on the outer level we optimize for . This permits us to give fast algorithms-exhibiting quadratic convergence-for solving the considered Hamiltonian matrix nearness problems.

Springer Heidelberg2015