Goal.Geometry.Differential.Convex

Tools are also provided for convex analysis, as the dual structures of convex analysis are equivalent to Riemannian manifolds with certain properties.

Documentation

class (Primal c, Manifold m) => Legendre c m where

Although convex analysis is usually developed seperately from differential geometry, it arrises naturally out of the theory of dually flat Manifolds.

A Manifold is Legendre for a particular coordinated system if it is associated with a particular convex function on points of the manifold known as a potential.

Methods

potential :: (c :#: m) -> Double

potentialDifferentials :: (c :#: m) -> Differentials :#: Tangent c m

Instances

Legendre c m => Legendre c (Replicated m)

potentialMapping :: Legendre c m => (c :#: m) -> Dual c :#: m

divergence :: (Primal c, Legendre c m, Legendre (Dual c) m) => (c :#: m) -> (Dual c :#: m) -> Double

Computes the divergence between two points.

legendreFlat :: (Legendre c m, Riemannian c m) => (c :#: m) -> (c :#: m) -> Dual c :#: m

Applies flat to the second input, based on the tangent space at the first input.