Goal.Geometry.Differential

This module provides tools for working with differential and Riemannian geometry.

Tangent Spaces

Types

newtype Tangent c m

Tangent spaces on Manifolds are the basis for differential geometry. Tangent spaces are defined at each point on a differentiable Manifold.

Constructors

Tangent

Fields

removeTangent :: c :#: m

Instances

(Manifold m, Riemannian c m) => Riemannian Partials (Tangent c m)

Eq m => Eq (Tangent c m)

Read m => Read (Tangent c m)

Show m => Show (Tangent c m)

Manifold m => Manifold (Tangent c m)

newtype Bundle c m

A Tangent Bundle is the original Manifold combined with all its Tangent spaces.

Constructors

Bundle

Fields

removeBundle :: m

Instances

Eq m => Eq (Bundle c m)

Read m => Read (Bundle c m)

Show m => Show (Bundle c m)

Manifold m => Manifold (Bundle c m)

data Partials

The Partials coordinate system is defined as the partial derivatives of the coordinate functions at a particular point.

Constructors

Partials

Instances

Eq Partials

Read Partials

Show Partials

Primal Partials

(Manifold m, Riemannian c m) => Riemannian Partials (Tangent c m)

type Dual Partials = Differentials

data Differentials

The Differentials coordinate system represents the set of linear functionals on the Tangent space.

Constructors

Differentials

Instances

Eq Differentials

Read Differentials

Show Differentials

Primal Differentials

type Dual Differentials = Partials

Functions

gradientStep :: Manifold m => Double -> (Partials :#: Tangent c m) -> c :#: m

gradientStep follows takes a gradient in a particular tangent space and transforms the point underlying the given tangent space by shifting it slightly in the direction of the gradient.

projectTangent :: (d :#: Tangent c m) -> c :#: m

Returns the underlying Point from a Tangent vector.

tangentToBundle :: Manifold m => (c :#: Tangent d m) -> c :#: Bundle d m

Converts a Tangent vector into a Point on a Tangent Bundle.

bundleToTangent :: Manifold m => (c :#: Bundle d m) -> c :#: Tangent d m

Converts a Point on a Tangent Bundle into a Tangent vector.

Riemannian Manifolds

class Manifold m => Riemannian c m where

Riemannian Manifolds are differentiable Manifolds where associated with each point in the Manifold is a Tangent space with a smoothly varying inner product. flat and sharp correspond to lowering and raising the indices via the musical isomorphism determined by the metric tensor.

A Riemannian Manifold should should satisfy the law

flat $ sharp p = p

Minimal complete definition

metric

Methods

metric :: (c :#: m) -> Function Partials Differentials :#: Tensor (Tangent c m) (Tangent c m)

flat :: (Partials :#: Tangent c m) -> Differentials :#: Tangent c m

sharp :: (Differentials :#: Tangent c m) -> Partials :#: Tangent c m

Instances

Riemannian Cartesian Continuum

Riemannian Cartesian Euclidean

(Manifold m, Riemannian c m) => Riemannian c (Replicated m)

(Manifold m, Riemannian c m) => Riemannian Partials (Tangent c m)

Gradient Pursuit

gradientAscent

Arguments

:: (Riemannian c m, Manifold m)

=> Double

Step size

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

Gradient calculator

-> (c :#: m)

The initial point

-> [c :#: m]

The gradient ascent

vanillaGradientAscent

Arguments

:: Manifold m

=> Double

Step size

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

Gradient calculator

-> (c :#: m)

The initial point

-> [c :#: m]

The gradient ascent

gradientDescent

Arguments

:: (Riemannian c m, Manifold m)

=> Double

Step size

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

Gradient calculator

-> (c :#: m)

The initial point

-> [c :#: m]

The gradient ascent

vanillaGradientDescent

Arguments

:: Manifold m

=> Double

Step size

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

Gradient calculator

-> (c :#: m)

The initial point

-> [c :#: m]

The gradient ascent