Data.Manifold.Types.Stiefel

Stiefel manifolds are a generalisation of the concept of the UnitSphere in real vector spaces. The n-th Stiefel manifold is the space of all possible configurations of n orthonormal vectors. In the case n = 1, simply a single normalised vector, i.e. a vector on the unit sphere.

Alternatively, the stiefel manifolds can be defined as quotient spaces under scalings, and we prefer that definition since it doesn't require a notion of unit length (which is only defined in inner-product spaces).

Documentation

newtype Stiefel1 v

Constructors

Stiefel1

Fields

• getStiefel1N :: DualVector v

Instances

Show (DualVector v) => Show (Stiefel1 v)

Defined in Data.Manifold.Types.Stiefel

Methods

showsPrec :: Int -> Stiefel1 v -> ShowS

show :: Stiefel1 v -> String

showList :: [Stiefel1 v] -> ShowS

(LinearSpace v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), StiefelScalar (Scalar v)) => Semimanifold (Stiefel1 v)

Defined in Data.Manifold.Types

Associated Types

type Needle (Stiefel1 v) :: Type

type Interior (Stiefel1 v) :: Type

Methods

(.+~^) :: Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Stiefel1 v

fromInterior :: Interior (Stiefel1 v) -> Stiefel1 v

toInterior :: Stiefel1 v -> Maybe (Interior (Stiefel1 v))

translateP :: Tagged (Stiefel1 v) (Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Interior (Stiefel1 v))

(.-~^) :: Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Stiefel1 v

semimanifoldWitness :: SemimanifoldWitness (Stiefel1 v)

(LinearSpace v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), StiefelScalar (Scalar v)) => PseudoAffine (Stiefel1 v)

Defined in Data.Manifold.Types

Methods

(.-~.) :: Stiefel1 v -> Stiefel1 v -> Maybe (Needle (Stiefel1 v))

(.-~!) :: Stiefel1 v -> Stiefel1 v -> Needle (Stiefel1 v)

pseudoAffineWitness :: PseudoAffineWitness (Stiefel1 v)

(Geodesic v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), LinearSpace v, Scalar v ~ ℝ, Geodesic (DualVector v), InnerSpace (DualVector v)) => Geodesic (Stiefel1 v)

Defined in Data.Manifold.Riemannian

Methods

geodesicBetween :: Stiefel1 v -> Stiefel1 v -> Maybe (D¹ -> Stiefel1 v)

geodesicWitness :: GeodesicWitness (Stiefel1 v)

middleBetween :: Stiefel1 v -> Stiefel1 v -> Maybe (Stiefel1 v)

type Interior (Stiefel1 v)

Defined in Data.Manifold.Types

type Interior (Stiefel1 v) = Stiefel1 v

type Needle (Stiefel1 v)

Defined in Data.Manifold.Types

type Needle (Stiefel1 v)