Copyright	(c) Justus Sagemüller 2016
License	GPL v3
Maintainer	(@) jsag $ hvl.no
Stability	experimental
Portability	portable
Safe Haskell	Safe-Inferred
Language	Haskell2010

Math.VectorSpace.ZeroDimensional

Description

Documentation

data ZeroDim s #

Constructors

Origin

Instances

Instances details

Dimensional 0 (ZeroDim s) Source #
Instance details Defined in Math.LinearMap.Category.Class Methods knownDimensionalitySing :: Sing 0 Source # unsafeFromArrayWithOffset :: Vector α (Scalar (ZeroDim s)) => Int -> α (Scalar (ZeroDim s)) -> ZeroDim s Source # unsafeWriteArrayWithOffset :: forall (α :: Type -> Type) σ. Vector α (Scalar (ZeroDim s)) => Mutable α σ (Scalar (ZeroDim s)) -> Int -> ZeroDim s -> ST σ () Source #
Monoid (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Methods mempty :: ZeroDim s # mappend :: ZeroDim s -> ZeroDim s -> ZeroDim s # mconcat :: [ZeroDim s] -> ZeroDim s #
Semigroup (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Methods (<>) :: ZeroDim s -> ZeroDim s -> ZeroDim s # sconcat :: NonEmpty (ZeroDim s) -> ZeroDim s # stimes :: Integral b => b -> ZeroDim s -> ZeroDim s #
Show (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Methods showsPrec :: Int -> ZeroDim s -> ShowS # show :: ZeroDim s -> String # showList :: [ZeroDim s] -> ShowS #
Eq (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Methods (==) :: ZeroDim s -> ZeroDim s -> Bool # (/=) :: ZeroDim s -> ZeroDim s -> Bool #
Num' s => LinearSpace (ZeroDim s) Source #
Instance details Defined in Math.LinearMap.Category.Class Associated Types type DualVector (ZeroDim s) Source # Methods dualSpaceWitness :: DualSpaceWitness (ZeroDim s) Source # linearId :: ZeroDim s +> ZeroDim s Source # idTensor :: ZeroDim s ⊗ DualVector (ZeroDim s) Source # sampleLinearFunction :: (TensorSpace w, Scalar (ZeroDim s) ~ Scalar w) => (ZeroDim s -+> w) -+> (ZeroDim s +> w) Source # toLinearForm :: DualVector (ZeroDim s) -+> (ZeroDim s +> Scalar (ZeroDim s)) Source # fromLinearForm :: (ZeroDim s +> Scalar (ZeroDim s)) -+> DualVector (ZeroDim s) Source # coerceDoubleDual :: VSCCoercion (Scalar (ZeroDim s)) (ZeroDim s) (DualVector (DualVector (ZeroDim s))) Source # trace :: (ZeroDim s +> ZeroDim s) -+> Scalar (ZeroDim s) Source # contractTensorMap :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s +> (ZeroDim s ⊗ w)) -+> w Source # contractMapTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ (ZeroDim s +> w)) -+> w Source # contractTensorFn :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s -+> (ZeroDim s ⊗ w)) -+> w Source # contractLinearMapAgainst :: (LinearSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (ZeroDim s +> w) (w -+> ZeroDim s) (Scalar (ZeroDim s)) Source # applyDualVector :: Bilinear (DualVector (ZeroDim s)) (ZeroDim s) (Scalar (ZeroDim s)) Source # applyLinear :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (ZeroDim s +> w) (ZeroDim s) w Source # composeLinear :: (LinearSpace w, TensorSpace x, Scalar w ~ Scalar (ZeroDim s), Scalar x ~ Scalar (ZeroDim s)) => Bilinear (w +> x) (ZeroDim s +> w) (ZeroDim s +> x) Source # tensorId :: (LinearSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) +> (ZeroDim s ⊗ w) Source # applyTensorFunctional :: (LinearSpace u, Scalar u ~ Scalar (ZeroDim s)) => Bilinear (DualVector (ZeroDim s ⊗ u)) (ZeroDim s ⊗ u) (Scalar (ZeroDim s)) Source # applyTensorLinMap :: (LinearSpace u, TensorSpace w, Scalar u ~ Scalar (ZeroDim s), Scalar w ~ Scalar (ZeroDim s)) => Bilinear ((ZeroDim s ⊗ u) +> w) (ZeroDim s ⊗ u) w Source # useTupleLinearSpaceComponents :: ZeroDim s ~ (x, y) => ((LinearSpace x, LinearSpace y, Scalar x ~ Scalar y) => φ) -> φ Source #
Num' s => TensorSpace (ZeroDim s) Source #
Instance details Defined in Math.LinearMap.Category.Class Associated Types type TensorProduct (ZeroDim s) w Source # Methods scalarSpaceWitness :: ScalarSpaceWitness (ZeroDim s) Source # linearManifoldWitness :: LinearManifoldWitness (ZeroDim s) Source # zeroTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => ZeroDim s ⊗ w Source # toFlatTensor :: ZeroDim s -+> (ZeroDim s ⊗ Scalar (ZeroDim s)) Source # fromFlatTensor :: (ZeroDim s ⊗ Scalar (ZeroDim s)) -+> ZeroDim s Source # addTensors :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w Source # subtractTensors :: (TensorSpace (ZeroDim s), TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w Source # scaleTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (Scalar (ZeroDim s)) (ZeroDim s ⊗ w) (ZeroDim s ⊗ w) Source # negateTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -+> (ZeroDim s ⊗ w) Source # tensorProduct :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (ZeroDim s) w (ZeroDim s ⊗ w) Source # tensorProducts :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => [(ZeroDim s, w)] -> ZeroDim s ⊗ w Source # transposeTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -+> (w ⊗ ZeroDim s) Source # fmapTensor :: (TensorSpace w, TensorSpace x, Scalar w ~ Scalar (ZeroDim s), Scalar x ~ Scalar (ZeroDim s)) => Bilinear (w -+> x) (ZeroDim s ⊗ w) (ZeroDim s ⊗ x) Source # fzipTensorWith :: (TensorSpace u, TensorSpace w, TensorSpace x, Scalar u ~ Scalar (ZeroDim s), Scalar w ~ Scalar (ZeroDim s), Scalar x ~ Scalar (ZeroDim s)) => Bilinear ((w, x) -+> u) (ZeroDim s ⊗ w, ZeroDim s ⊗ x) (ZeroDim s ⊗ u) Source # tensorUnsafeFromArrayWithOffset :: forall w α (n :: Nat) (m :: Nat). (Dimensional n (ZeroDim s), TensorSpace w, Dimensional m w, Scalar w ~ Scalar (ZeroDim s), Vector α (Scalar (ZeroDim s))) => Int -> α (Scalar (ZeroDim s)) -> ZeroDim s ⊗ w Source # tensorUnsafeWriteArrayWithOffset :: forall w (α :: Type -> Type) σ (n :: Nat) (m :: Nat). (Dimensional n (ZeroDim s), TensorSpace w, Dimensional m w, Scalar w ~ Scalar (ZeroDim s), Vector α (Scalar (ZeroDim s))) => Mutable α σ (Scalar (ZeroDim s)) -> Int -> (ZeroDim s ⊗ w) -> ST σ () Source # coerceFmapTensorProduct :: (Functor p, TensorSpace a, Scalar a ~ Scalar (ZeroDim s), TensorSpace b, Scalar b ~ Scalar (ZeroDim s)) => p (ZeroDim s) -> VSCCoercion (Scalar (ZeroDim s)) a b -> Coercion (TensorProduct (ZeroDim s) a) (TensorProduct (ZeroDim s) b) Source # wellDefinedVector :: ZeroDim s -> Maybe (ZeroDim s) Source # wellDefinedTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> Maybe (ZeroDim s ⊗ w) Source #
DimensionAware (ZeroDim s) Source #
Instance details Defined in Math.LinearMap.Category.Class Associated Types type StaticDimension (ZeroDim s) :: Maybe Nat Source # Methods dimensionalityWitness :: DimensionalityWitness (ZeroDim s) Source #
Num' s => FiniteDimensional (ZeroDim s) Source #
Instance details Defined in Math.VectorSpace.Docile Associated Types data SubBasis (ZeroDim s) Source # Methods entireBasis :: SubBasis (ZeroDim s) Source # enumerateSubBasis :: SubBasis (ZeroDim s) -> [ZeroDim s] Source # subbasisDimension :: SubBasis (ZeroDim s) -> Int Source # decomposeLinMap :: (LSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s +> w) -> (SubBasis (ZeroDim s), DList w) Source # decomposeLinMapWithin :: (LSpace w, Scalar w ~ Scalar (ZeroDim s)) => SubBasis (ZeroDim s) -> (ZeroDim s +> w) -> Either (SubBasis (ZeroDim s), DList w) (DList w) Source # recomposeSB :: SubBasis (ZeroDim s) -> [Scalar (ZeroDim s)] -> (ZeroDim s, [Scalar (ZeroDim s)]) Source # recomposeSBTensor :: (FiniteDimensional w, Scalar w ~ Scalar (ZeroDim s)) => SubBasis (ZeroDim s) -> SubBasis w -> [Scalar (ZeroDim s)] -> (ZeroDim s ⊗ w, [Scalar (ZeroDim s)]) Source # recomposeLinMap :: (LSpace w, Scalar w ~ Scalar (ZeroDim s)) => SubBasis (ZeroDim s) -> [w] -> (ZeroDim s +> w, [w]) Source # recomposeContraLinMap :: (LinearSpace w, Scalar w ~ Scalar (ZeroDim s), Functor f) => (f (Scalar w) -> w) -> f (DualVector (ZeroDim s)) -> ZeroDim s +> w Source # recomposeContraLinMapTensor :: (FiniteDimensional u, LinearSpace w, Scalar u ~ Scalar (ZeroDim s), Scalar w ~ Scalar (ZeroDim s), Functor f) => (f (Scalar w) -> w) -> f (ZeroDim s +> DualVector u) -> (ZeroDim s ⊗ u) +> w Source # uncanonicallyFromDual :: DualVector (ZeroDim s) -+> ZeroDim s Source # uncanonicallyToDual :: ZeroDim s -+> DualVector (ZeroDim s) Source # tensorEquality :: (TensorSpace w, Eq w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> Bool Source # dualFinitenessWitness :: DualFinitenessWitness (ZeroDim s) Source #
(Fractional' s, SemiInner s) => SemiInner (ZeroDim s) Source #
Instance details Defined in Math.VectorSpace.Docile Methods dualBasisCandidates :: [(Int, ZeroDim s)] -> Forest (Int, DualVector (ZeroDim s)) Source # tensorDualBasisCandidates :: (SemiInner w, Scalar w ~ Scalar (ZeroDim s)) => [(Int, ZeroDim s ⊗ w)] -> Forest (Int, DualVector (ZeroDim s ⊗ w)) Source # symTensorDualBasisCandidates :: [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s))] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (DualVector (ZeroDim s))) Source # symTensorTensorDualBasisCandidates :: (SemiInner w, Scalar w ~ Scalar (ZeroDim s)) => [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s) ⊗ w)] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s) +> DualVector w) Source #
PseudoAffine (ZeroDim k)
Instance details Defined in Math.Manifold.Core.PseudoAffine Methods (.-~.) :: ZeroDim k -> ZeroDim k -> Maybe (Needle (ZeroDim k)) # (.-~!) :: ZeroDim k -> ZeroDim k -> Needle (ZeroDim k) # pseudoAffineWitness :: PseudoAffineWitness (ZeroDim k) #
Semimanifold (ZeroDim k)
Instance details Defined in Math.Manifold.Core.PseudoAffine Associated Types type Needle (ZeroDim k) # Methods (.+~^) :: ZeroDim k -> Needle (ZeroDim k) -> ZeroDim k # (.-~^) :: ZeroDim k -> Needle (ZeroDim k) -> ZeroDim k # semimanifoldWitness :: SemimanifoldWitness (ZeroDim k) #
AdditiveGroup (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Methods zeroV :: ZeroDim s # (^+^) :: ZeroDim s -> ZeroDim s -> ZeroDim s # negateV :: ZeroDim s -> ZeroDim s # (^-^) :: ZeroDim s -> ZeroDim s -> ZeroDim s #
AffineSpace (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Associated Types type Diff (ZeroDim s) # Methods (.-.) :: ZeroDim s -> ZeroDim s -> Diff (ZeroDim s) # (.+^) :: ZeroDim s -> Diff (ZeroDim s) -> ZeroDim s #
HasBasis (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Associated Types type Basis (ZeroDim s) # Methods basisValue :: Basis (ZeroDim s) -> ZeroDim s # decompose :: ZeroDim s -> [(Basis (ZeroDim s), Scalar (ZeroDim s))] # decompose' :: ZeroDim s -> Basis (ZeroDim s) -> Scalar (ZeroDim s) #
AdditiveGroup s => InnerSpace (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Methods (<.>) :: ZeroDim s -> ZeroDim s -> Scalar (ZeroDim s) #
VectorSpace (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional Associated Types type Scalar (ZeroDim s) # Methods (*^) :: Scalar (ZeroDim s) -> ZeroDim s -> ZeroDim s #
Show (LinearMap s v (ZeroDim s)) Source #
Instance details Defined in Math.VectorSpace.Docile Methods showsPrec :: Int -> LinearMap s v (ZeroDim s) -> ShowS # show :: LinearMap s v (ZeroDim s) -> String # showList :: [LinearMap s v (ZeroDim s)] -> ShowS #
type DualVector (ZeroDim s) Source #
Instance details Defined in Math.LinearMap.Category.Class type DualVector (ZeroDim s) = ZeroDim s
type StaticDimension (ZeroDim s) Source #
Instance details Defined in Math.LinearMap.Category.Class type StaticDimension (ZeroDim s) = 'Just 0
data SubBasis (ZeroDim s) Source #
Instance details Defined in Math.VectorSpace.Docile data SubBasis (ZeroDim s) = ZeroBasis
type Needle (ZeroDim k)
Instance details Defined in Math.Manifold.Core.PseudoAffine type Needle (ZeroDim k) = ZeroDim k
type Diff (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional type Diff (ZeroDim s) = ZeroDim s
type Basis (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional type Basis (ZeroDim s) = Void
type Scalar (ZeroDim s)
Instance details Defined in Math.Manifold.VectorSpace.ZeroDimensional type Scalar (ZeroDim s) = s
type TensorProduct (ZeroDim s) v Source #
Instance details Defined in Math.LinearMap.Category.Class type TensorProduct (ZeroDim s) v = ZeroDim s