Downhill.Grad

Documentation

class (Scalar v ~ Scalar dv, AdditiveGroup (Scalar v), VectorSpace v, VectorSpace dv) => Dual v dv where

Dual of a vector v is a linear map v -> Scalar v.

Minimal complete definition

Nothing

Methods

evalGrad :: dv -> v -> Scalar v

default evalGrad :: (GDual (Scalar v) (Rep v) (Rep dv), Generic dv, Generic v) => dv -> v -> Scalar v

Instances

Dual Integer Integer

Defined in Downhill.Grad

Methods

evalGrad :: Integer -> Integer -> Scalar Integer

Dual Double Double

Defined in Downhill.Grad

Methods

evalGrad :: Double -> Double -> Scalar Double

Dual Float Float

Defined in Downhill.Grad

Methods

evalGrad :: Float -> Float -> Scalar Float

Num a => Dual (AsNum a) (AsNum a)

Defined in Downhill.BVar.Num

Methods

evalGrad :: AsNum a -> AsNum a -> Scalar (AsNum a)

(HasGrad (Scalar v), HasGrad v, HasGrad dv, Dual v dv, Grad dv ~ v, Grad v ~ dv, Tang v ~ v, Tang dv ~ dv, Grad (Scalar dv) ~ Scalar dv) => Dual (BVar r v) (BVar r dv)

Defined in Downhill.BVar

Methods

evalGrad :: BVar r dv -> BVar r v -> Scalar (BVar r v)

(Scalar a ~ Scalar b, Dual a da, Dual b db) => Dual (a, b) (da, db)

Defined in Downhill.Grad

Methods

evalGrad :: (da, db) -> (a, b) -> Scalar (a, b)

(Scalar a ~ Scalar b, Scalar a ~ Scalar c, Dual a da, Dual b db, Dual c dc) => Dual (a, b, c) (da, db, dc)

Defined in Downhill.Grad

Methods

evalGrad :: (da, db, dc) -> (a, b, c) -> Scalar (a, b, c)

class Dual v dv => HilbertSpace v dv where

u . v = evalDual (riesz u) v | du . dv = evalDual du (coriesz dv)

Methods

riesz :: v -> dv

coriesz :: dv -> v

Instances

(HilbertSpace v dv, HasGrad (Scalar v), HasGrad v, HasGrad dv, Grad dv ~ v, Grad v ~ dv, Tang v ~ v, Tang dv ~ dv, Grad (Scalar dv) ~ Scalar dv) => HilbertSpace (BVar r v) (BVar r dv)

Defined in Downhill.BVar

Methods

riesz :: BVar r v -> BVar r dv

coriesz :: BVar r dv -> BVar r v

class (Dual (Tang p) (Grad p), Scalar (Tang p) ~ Scalar (Grad p)) => Manifold p

Associated Types

type Tang p :: Type

Tangent space.

type Grad p :: Type

Cotangent space.

Instances

Manifold Integer

Defined in Downhill.Grad

Associated Types

type Tang Integer

type Grad Integer

Manifold Double

Defined in Downhill.Grad

Associated Types

type Tang Double

type Grad Double

Manifold Float

Defined in Downhill.Grad

Associated Types

type Tang Float

type Grad Float

Num a => Manifold (AsNum a)

Defined in Downhill.BVar.Num

Associated Types

type Tang (AsNum a)

type Grad (AsNum a)

(HasGrad (MScalar p), HasGrad (Tang p), HasGrad (Grad p), Grad (Grad p) ~ Tang p, Tang (Grad p) ~ Grad p, Tang (Tang p) ~ Tang p, Grad (Tang p) ~ Grad p, Grad (MScalar p) ~ MScalar p, Scalar (Grad p) ~ Scalar (Tang p), Manifold p) => Manifold (BVar r p)

Defined in Downhill.BVar

Associated Types

type Tang (BVar r p)

type Grad (BVar r p)

Manifold a => Manifold (TraversableVar f a)

Defined in Downhill.BVar.Traversable

Associated Types

type Tang (TraversableVar f a)

type Grad (TraversableVar f a)

(HasGrad a, HasGrad b, MScalar b ~ MScalar a) => Manifold (a, b)

Defined in Downhill.Grad

Associated Types

type Tang (a, b)

type Grad (a, b)

(HasGrad a, HasGrad b, HasGrad c, MScalar b ~ MScalar a, MScalar c ~ MScalar a) => Manifold (a, b, c)

Defined in Downhill.Grad

Associated Types

type Tang (a, b, c)

type Grad (a, b, c)

type HasGrad p = (Manifold p, BasicVector (Grad p))

Differentiable functions don't need to be constrained to vector spaces, they can be defined on other smooth manifolds, too.

type MScalar p = Scalar (Tang p)

type GradBuilder v = VecBuilder (Grad v)

type HasGradAffine p = (AffineSpace p, HasGrad p, HasGrad (Tang p), Tang p ~ Diff p, Tang (Tang p) ~ Tang p, Grad (Tang p) ~ Grad p)