Safe Haskell	Safe-Inferred
Language	Haskell2010

Algebra.Linear

Synopsis

type VectorSpace scalar a = (Field scalar, Module scalar a, Group a)
class (Finite (Rep v), Representable v, Foldable v, Applicative v) => VectorR v where
- vectorSplit :: v ~ (f ⊗ g) => Dict (VectorR f, VectorR g)
- vectorCut :: v ~ (f ⊕ g) => Dict (VectorR f, VectorR g)
class VectorR v => InnerProdSpace v where
- inner :: Field s => v s -> v s -> s
type VZero x = Zero x
data VNext v a = VNext {
- vnextInit :: !(v a)
- vnextLast :: !a
}
data V f a where
- V0 :: V One a
- (:/) :: !(V f a) -> !a -> V (VNext f) a
class (Foldable f, Applicative f) => IsVec f where
- reifyVec :: f a -> V f a
fromV :: V f a -> f a
type V1' = VNext One
type V2' = VNext V1'
type V3' = VNext V2'
pattern V1' :: a -> V1' a
pattern V2' :: forall a. a -> a -> V2' a
pattern V3' :: forall a. a -> a -> a -> V3' a
newtype Euclid f a = Euclid {
- fromEuclid :: f a
}
type V3 = Euclid V3'
type V2 = Euclid V2'
pattern V2 :: forall a. a -> a -> Euclid V2' a
pattern V3 :: forall a. a -> a -> a -> Euclid V3' a
pureMat :: (Applicative v, Applicative w) => s -> Mat s v w
(⊙) :: Applicative v => Multiplicative s => v s -> v s -> v s
(·) :: Field s => InnerProdSpace v => v s -> v s -> s
sqNorm :: Field s => InnerProdSpace v => v s -> s
norm :: Algebraic s => InnerProdSpace v => v s -> s
normalize :: VectorSpace s (v s) => Algebraic s => InnerProdSpace v => v s -> v s
(×) :: Ring a => V3 a -> V3 a -> V3 a
type SqMat v s = Mat s v v
newtype Mat s w v = Mat {
- fromMat :: w (v s)
}
newtype Flat w v s = Flat {
- fromFlat :: w (v s)
}
flatMat :: Mat s w v -> Flat w v s
matFlat :: Flat w v s -> Mat s w v
fromRel :: (VectorR a, VectorR b) => Rel s (Rep a) (Rep b) -> Mat s a b
type Mat3x3 s = SqMat V3 s
type Mat2x2 s = SqMat V2 s
pattern Mat2x2 :: forall s. s -> s -> s -> s -> Mat s V2 V2
pattern Mat3x3 :: forall s. s -> s -> s -> s -> s -> s -> s -> s -> s -> Mat s V3 V3
rotation2d :: Transcendental a => a -> Mat2x2 a
crossProductMatrix :: Group a => V3 a -> Mat3x3 a
outerWith :: (Applicative v, Applicative w) => (s -> t -> u) -> w s -> v t -> Mat u v w
outer :: (Applicative v, Applicative w, Multiplicative s) => Euclid w s -> Euclid v s -> Mat s (Euclid w) (Euclid v)
diagonal :: Eq (Rep v) => Representable v => Ring s => Applicative v => v s -> SqMat v s
rotation3d :: Transcendental a => a -> V3 a -> Mat3x3 a
rotationFromTo :: forall a. Algebraic a => V3 a -> V3 a -> Mat3x3 a
transpose :: Functor f => Distributive g => Mat a f g -> Mat a g f
matMul :: (Foldable u, Ring s, Applicative w, Applicative v, Applicative u) => Mat s u w -> Mat s v u -> Mat s v w
(<+>) :: (Applicative f, Additive b) => f b -> f b -> f b
matVecMul :: forall s v w. (Ring s, Foldable v, Applicative v, Applicative w) => Mat s v w -> v s -> w s
prop_linear_with_functor_laws :: Property
runTests :: IO Bool

Documentation

type VectorSpace scalar a = (Field scalar, Module scalar a, Group a) Source #

class (Finite (Rep v), Representable v, Foldable v, Applicative v) => VectorR v where Source #

Representation of vector as traversable functor ... but this is missing the link with *^ for module. We should be able to add forall s. PreRing s => Module s (v s), but this creates problems when defining instances.

Minimal complete definition

Nothing

Methods

vectorSplit :: v ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source #

vectorCut :: v ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #

Instances

Instances details

VectorR Id Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). Id ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). Id ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
VectorR a => VectorR (VNext a) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). VNext a ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). VNext a ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
VectorR (One :: Type -> Type) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). One ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). One ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
VectorR f => VectorR (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f0 :: Type -> Type) (g :: Type -> Type). Euclid f ~ (f0 ⊗ g) => Dict (VectorR f0, VectorR g) Source # vectorCut :: forall (f0 :: Type -> Type) (g :: Type -> Type). Euclid f ~ (f0 ⊕ g) => Dict (VectorR f0, VectorR g) Source #
(VectorR v, VectorR w) => VectorR (v ⊗ w) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). (v ⊗ w) ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). (v ⊗ w) ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
(VectorR v, VectorR w) => VectorR (v ∘ w) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). (v ∘ w) ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). (v ∘ w) ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #

class VectorR v => InnerProdSpace v where Source #

Methods

inner :: Field s => v s -> v s -> s Source #

Instances

Instances details

VectorR f => InnerProdSpace (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods inner :: Field s => Euclid f s -> Euclid f s -> s Source #

type VZero x = Zero x Source #

data VNext v a Source #

Constructors

VNext
Fields vnextInit :: !(v a) vnextLast :: !a

Instances

Instances details

Representable a => Representable (VNext a) Source #
Instance details Defined in Algebra.Linear Associated Types type Rep (VNext a) # Methods tabulate :: (Rep (VNext a) -> a0) -> VNext a a0 # index :: VNext a a0 -> Rep (VNext a) -> a0 #
Foldable v => Foldable (VNext v) Source #
Instance details Defined in Algebra.Linear Methods fold :: Monoid m => VNext v m -> m # foldMap :: Monoid m => (a -> m) -> VNext v a -> m # foldMap' :: Monoid m => (a -> m) -> VNext v a -> m # foldr :: (a -> b -> b) -> b -> VNext v a -> b # foldr' :: (a -> b -> b) -> b -> VNext v a -> b # foldl :: (b -> a -> b) -> b -> VNext v a -> b # foldl' :: (b -> a -> b) -> b -> VNext v a -> b # foldr1 :: (a -> a -> a) -> VNext v a -> a # foldl1 :: (a -> a -> a) -> VNext v a -> a # toList :: VNext v a -> [a] # null :: VNext v a -> Bool # length :: VNext v a -> Int # elem :: Eq a => a -> VNext v a -> Bool # maximum :: Ord a => VNext v a -> a # minimum :: Ord a => VNext v a -> a # sum :: Num a => VNext v a -> a # product :: Num a => VNext v a -> a #
Traversable v => Traversable (VNext v) Source #
Instance details Defined in Algebra.Linear Methods traverse :: Applicative f => (a -> f b) -> VNext v a -> f (VNext v b) # sequenceA :: Applicative f => VNext v (f a) -> f (VNext v a) # mapM :: Monad m => (a -> m b) -> VNext v a -> m (VNext v b) # sequence :: Monad m => VNext v (m a) -> m (VNext v a) #
Applicative v => Applicative (VNext v) Source #
Instance details Defined in Algebra.Linear Methods pure :: a -> VNext v a # (<>) :: VNext v (a -> b) -> VNext v a -> VNext v b # liftA2 :: (a -> b -> c) -> VNext v a -> VNext v b -> VNext v c # (>) :: VNext v a -> VNext v b -> VNext v b # (<*) :: VNext v a -> VNext v b -> VNext v a #
Functor v => Functor (VNext v) Source #
Instance details Defined in Algebra.Linear Methods fmap :: (a -> b) -> VNext v a -> VNext v b # (<$) :: a -> VNext v b -> VNext v a #
Distributive v => Distributive (VNext v) Source #
Instance details Defined in Algebra.Linear Methods distribute :: Functor f => f (VNext v a) -> VNext v (f a) # collect :: Functor f => (a -> VNext v b) -> f a -> VNext v (f b) # distributeM :: Monad m => m (VNext v a) -> VNext v (m a) # collectM :: Monad m => (a -> VNext v b) -> m a -> VNext v (m b) #
IsVec f => IsVec (VNext f) Source #
Instance details Defined in Algebra.Linear Methods reifyVec :: VNext f a -> V (VNext f) a Source #
VectorR a => VectorR (VNext a) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). VNext a ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). VNext a ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
(Show a, Show (v a)) => Show (VNext v a) Source #
Instance details Defined in Algebra.Linear Methods showsPrec :: Int -> VNext v a -> ShowS # show :: VNext v a -> String # showList :: [VNext v a] -> ShowS #
(Eq a, Eq (v a)) => Eq (VNext v a) Source #
Instance details Defined in Algebra.Linear Methods (==) :: VNext v a -> VNext v a -> Bool # (/=) :: VNext v a -> VNext v a -> Bool #
(Ord a, Ord (v a)) => Ord (VNext v a) Source #
Instance details Defined in Algebra.Linear Methods compare :: VNext v a -> VNext v a -> Ordering # (<) :: VNext v a -> VNext v a -> Bool # (<=) :: VNext v a -> VNext v a -> Bool # (>) :: VNext v a -> VNext v a -> Bool # (>=) :: VNext v a -> VNext v a -> Bool # max :: VNext v a -> VNext v a -> VNext v a # min :: VNext v a -> VNext v a -> VNext v a #
type Rep (VNext a) Source #
Instance details Defined in Algebra.Linear type Rep (VNext a) = (One :: Type) ⊕ Rep a

data V f a where Source #

Constructors

V0 :: V One a
(:/) :: !(V f a) -> !a -> V (VNext f) a

Instances

Instances details

Foldable (V f) Source #
Instance details Defined in Algebra.Linear Methods fold :: Monoid m => V f m -> m # foldMap :: Monoid m => (a -> m) -> V f a -> m # foldMap' :: Monoid m => (a -> m) -> V f a -> m # foldr :: (a -> b -> b) -> b -> V f a -> b # foldr' :: (a -> b -> b) -> b -> V f a -> b # foldl :: (b -> a -> b) -> b -> V f a -> b # foldl' :: (b -> a -> b) -> b -> V f a -> b # foldr1 :: (a -> a -> a) -> V f a -> a # foldl1 :: (a -> a -> a) -> V f a -> a # toList :: V f a -> [a] # null :: V f a -> Bool # length :: V f a -> Int # elem :: Eq a => a -> V f a -> Bool # maximum :: Ord a => V f a -> a # minimum :: Ord a => V f a -> a # sum :: Num a => V f a -> a # product :: Num a => V f a -> a #
Traversable (V f) Source #
Instance details Defined in Algebra.Linear Methods traverse :: Applicative f0 => (a -> f0 b) -> V f a -> f0 (V f b) # sequenceA :: Applicative f0 => V f (f0 a) -> f0 (V f a) # mapM :: Monad m => (a -> m b) -> V f a -> m (V f b) # sequence :: Monad m => V f (m a) -> m (V f a) #
IsVec f => Applicative (V f) Source #
Instance details Defined in Algebra.Linear Methods pure :: a -> V f a # (<>) :: V f (a -> b) -> V f a -> V f b # liftA2 :: (a -> b -> c) -> V f a -> V f b -> V f c # (>) :: V f a -> V f b -> V f b # (<*) :: V f a -> V f b -> V f a #
Functor (V f) Source #
Instance details Defined in Algebra.Linear Methods fmap :: (a -> b) -> V f a -> V f b # (<$) :: a -> V f b -> V f a #
Show a => Show (V f a) Source #
Instance details Defined in Algebra.Linear Methods showsPrec :: Int -> V f a -> ShowS # show :: V f a -> String # showList :: [V f a] -> ShowS #
Eq a => Eq (V f a) Source #
Instance details Defined in Algebra.Linear Methods (==) :: V f a -> V f a -> Bool # (/=) :: V f a -> V f a -> Bool #

class (Foldable f, Applicative f) => IsVec f where Source #

Methods

reifyVec :: f a -> V f a Source #

Instances

Instances details

IsVec f => IsVec (VNext f) Source #
Instance details Defined in Algebra.Linear Methods reifyVec :: VNext f a -> V (VNext f) a Source #
IsVec (One :: Type -> Type) Source #
Instance details Defined in Algebra.Linear Methods reifyVec :: One a -> V One a Source #

fromV :: V f a -> f a Source #

type V1' = VNext One Source #

type V2' = VNext V1' Source #

type V3' = VNext V2' Source #

pattern V1' :: a -> V1' a Source #

pattern V2' :: forall a. a -> a -> V2' a Source #

pattern V3' :: forall a. a -> a -> a -> V3' a Source #

newtype Euclid f a Source #

Make a Euclidean vector out of a traversable functor. (The p)

Constructors

Euclid
Fields fromEuclid :: f a

Instances

Instances details

(Functor f, Scalable s a) => Scalable s (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Euclid f a -> Euclid f a Source #
Representable f => Representable (Euclid f) Source #
Instance details Defined in Algebra.Linear Associated Types type Rep (Euclid f) # Methods tabulate :: (Rep (Euclid f) -> a) -> Euclid f a # index :: Euclid f a -> Rep (Euclid f) -> a #
Foldable f => Foldable (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods fold :: Monoid m => Euclid f m -> m # foldMap :: Monoid m => (a -> m) -> Euclid f a -> m # foldMap' :: Monoid m => (a -> m) -> Euclid f a -> m # foldr :: (a -> b -> b) -> b -> Euclid f a -> b # foldr' :: (a -> b -> b) -> b -> Euclid f a -> b # foldl :: (b -> a -> b) -> b -> Euclid f a -> b # foldl' :: (b -> a -> b) -> b -> Euclid f a -> b # foldr1 :: (a -> a -> a) -> Euclid f a -> a # foldl1 :: (a -> a -> a) -> Euclid f a -> a # toList :: Euclid f a -> [a] # null :: Euclid f a -> Bool # length :: Euclid f a -> Int # elem :: Eq a => a -> Euclid f a -> Bool # maximum :: Ord a => Euclid f a -> a # minimum :: Ord a => Euclid f a -> a # sum :: Num a => Euclid f a -> a # product :: Num a => Euclid f a -> a #
Traversable f => Traversable (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods traverse :: Applicative f0 => (a -> f0 b) -> Euclid f a -> f0 (Euclid f b) # sequenceA :: Applicative f0 => Euclid f (f0 a) -> f0 (Euclid f a) # mapM :: Monad m => (a -> m b) -> Euclid f a -> m (Euclid f b) # sequence :: Monad m => Euclid f (m a) -> m (Euclid f a) #
Applicative f => Applicative (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods pure :: a -> Euclid f a # (<>) :: Euclid f (a -> b) -> Euclid f a -> Euclid f b # liftA2 :: (a -> b -> c) -> Euclid f a -> Euclid f b -> Euclid f c # (>) :: Euclid f a -> Euclid f b -> Euclid f b # (<*) :: Euclid f a -> Euclid f b -> Euclid f a #
Functor f => Functor (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods fmap :: (a -> b) -> Euclid f a -> Euclid f b # (<$) :: a -> Euclid f b -> Euclid f a #
Distributive f => Distributive (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods distribute :: Functor f0 => f0 (Euclid f a) -> Euclid f (f0 a) # collect :: Functor f0 => (a -> Euclid f b) -> f0 a -> Euclid f (f0 b) # distributeM :: Monad m => m (Euclid f a) -> Euclid f (m a) # collectM :: Monad m => (a -> Euclid f b) -> m a -> Euclid f (m b) #
VectorR f => InnerProdSpace (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods inner :: Field s => Euclid f s -> Euclid f s -> s Source #
VectorR f => VectorR (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f0 :: Type -> Type) (g :: Type -> Type). Euclid f ~ (f0 ⊗ g) => Dict (VectorR f0, VectorR g) Source # vectorCut :: forall (f0 :: Type -> Type) (g :: Type -> Type). Euclid f ~ (f0 ⊕ g) => Dict (VectorR f0, VectorR g) Source #
Show (f a) => Show (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods showsPrec :: Int -> Euclid f a -> ShowS # show :: Euclid f a -> String # showList :: [Euclid f a] -> ShowS #
(Applicative f, Additive a) => Additive (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (+) :: Euclid f a -> Euclid f a -> Euclid f a Source # zero :: Euclid f a Source # times :: Natural -> Euclid f a -> Euclid f a Source #
(Applicative f, Group a) => Group (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (-) :: Euclid f a -> Euclid f a -> Euclid f a Source # subtract :: Euclid f a -> Euclid f a -> Euclid f a Source # negate :: Euclid f a -> Euclid f a Source # mult :: Integer -> Euclid f a -> Euclid f a Source #
Eq (f a) => Eq (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (==) :: Euclid f a -> Euclid f a -> Bool # (/=) :: Euclid f a -> Euclid f a -> Bool #
Ord (f a) => Ord (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods compare :: Euclid f a -> Euclid f a -> Ordering # (<) :: Euclid f a -> Euclid f a -> Bool # (<=) :: Euclid f a -> Euclid f a -> Bool # (>) :: Euclid f a -> Euclid f a -> Bool # (>=) :: Euclid f a -> Euclid f a -> Bool # max :: Euclid f a -> Euclid f a -> Euclid f a # min :: Euclid f a -> Euclid f a -> Euclid f a #
type Rep (Euclid f) Source #
Instance details Defined in Algebra.Linear type Rep (Euclid f) = Rep f

type V3 = Euclid V3' Source #

type V2 = Euclid V2' Source #

pattern V2 :: forall a. a -> a -> Euclid V2' a Source #

pattern V3 :: forall a. a -> a -> a -> Euclid V3' a Source #

pureMat :: (Applicative v, Applicative w) => s -> Mat s v w Source #

(⊙) :: Applicative v => Multiplicative s => v s -> v s -> v s Source #

Hadamard product

(·) :: Field s => InnerProdSpace v => v s -> v s -> s Source #

sqNorm :: Field s => InnerProdSpace v => v s -> s Source #

norm :: Algebraic s => InnerProdSpace v => v s -> s Source #

normalize :: VectorSpace s (v s) => Algebraic s => InnerProdSpace v => v s -> v s Source #

(×) :: Ring a => V3 a -> V3 a -> V3 a Source #

Cross product in 3 dimensions https://en.wikipedia.org/wiki/Cross_product

type SqMat v s = Mat s v v Source #

newtype Mat s w v Source #

Matrix type. (w s) is a column. (v s) is a row.

Constructors

Mat
Fields fromMat :: w (v s)

Instances

Instances details

(Functor f, Functor g, Scalable s a) => Scalable s (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Mat a f g -> Mat a f g Source #
Ring s => Braided ((⊗) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) (One :: Type -> Type) (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods swap :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ⊗ b) (b ⊗ a) Source # swap_ :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ⊗ b) (b ⊗ a) Source #
Ring s => Cartesian ((⊗) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) (One :: Type -> Type) (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods exl :: forall (a :: k) (b :: k). O2 (Mat s) a b => Mat s (a ⊗ b) a Source # exr :: forall (a :: k) (b :: k). O2 (Mat s) a b => Mat s (a ⊗ b) b Source # dis :: forall (a :: k). Obj (Mat s) a => Mat s a One Source # dup :: forall (a :: k). Obj (Mat s) a => Mat s a (a ⊗ a) Source # (▵) :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s a b -> Mat s a c -> Mat s a (b ⊗ c) Source #
Ring s => CoCartesian ((⊗) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) (One :: Type -> Type) (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods inl :: forall (a :: k) (b :: k). O2 (Mat s) a b => Mat s a (a ⊗ b) Source # inr :: forall (a :: k) (b :: k). O2 (Mat s) a b => Mat s b (a ⊗ b) Source # new :: forall (a :: k). Obj (Mat s) a => Mat s One a Source # jam :: forall (a :: k). Obj (Mat s) a => Mat s (a ⊗ a) a Source # (▿) :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s b a -> Mat s c a -> Mat s (b ⊗ c) a Source #
Ring s => Monoidal ((⊗) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) (One :: Type -> Type) (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods (⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c, Obj (Mat s) d) => Mat s a b -> Mat s c d -> Mat s (a ⊗ c) (b ⊗ d) Source # assoc :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s ((a ⊗ b) ⊗ c) (a ⊗ (b ⊗ c)) Source # assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s (a ⊗ (b ⊗ c)) ((a ⊗ b) ⊗ c) Source # unitorR :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) One) => Mat s a (a ⊗ One) Source # unitorR_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) One) => Mat s (a ⊗ One) a Source # unitorL :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) One) => Mat s a (One ⊗ a) Source # unitorL_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) One) => Mat s (One ⊗ a) a Source #
Ring s => Symmetric ((⊗) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) (One :: Type -> Type) (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear
Ring s => Braided ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods swap :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a) Source # swap_ :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a) Source #
Ring s => Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods (⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c, Obj (Mat s) d) => Mat s a b -> Mat s c d -> Mat s (a ∘ c) (b ∘ d) Source # assoc :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s ((a ∘ b) ∘ c) (a ∘ (b ∘ c)) Source # assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s (a ∘ (b ∘ c)) ((a ∘ b) ∘ c) Source # unitorR :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (a ∘ Id) Source # unitorR_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (a ∘ Id) a Source # unitorL :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (Id ∘ a) Source # unitorL_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (Id ∘ a) a Source #
Ring s => Symmetric ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear
Ring s => Category (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Associated Types type Obj (Mat s) :: k -> Constraint Source # Methods (.) :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s b c -> Mat s a b -> Mat s a c Source # id :: forall (a :: k). Obj (Mat s) a => Mat s a a Source #
Ring s => Dagger (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Methods dagger :: forall (a :: k) (b :: k). O2 (Mat s) a b => Mat s a b -> Mat s b a Source #
(Arbitrary s, Arbitrary1 a, Arbitrary1 b) => Arbitrary (Mat s a b) Source #
Instance details Defined in Algebra.Linear Methods arbitrary :: Gen (Mat s a b) # shrink :: Mat s a b -> [Mat s a b] #
Show (w (v s)) => Show (Mat s w v) Source #
Instance details Defined in Algebra.Linear Methods showsPrec :: Int -> Mat s w v -> ShowS # show :: Mat s w v -> String # showList :: [Mat s w v] -> ShowS #
(Applicative f, Applicative g, AbelianAdditive a) => AbelianAdditive (Mat a f g) Source #
Instance details Defined in Algebra.Linear
(Applicative f, Applicative g, Additive a) => Additive (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (+) :: Mat a f g -> Mat a f g -> Mat a f g Source # zero :: Mat a f g Source # times :: Natural -> Mat a f g -> Mat a f g Source #
(Applicative f, Applicative g, Group a) => Group (Mat a f g) Source #
Instance details Defined in Algebra.Linear Methods (-) :: Mat a f g -> Mat a f g -> Mat a f g Source # subtract :: Mat a f g -> Mat a f g -> Mat a f g Source # negate :: Mat a f g -> Mat a f g Source # mult :: Integer -> Mat a f g -> Mat a f g Source #
(TestEqual s, Arbitrary s, Arbitrary1 a, Arbitrary1 b, Show (a (b s)), VectorR b, VectorR a) => TestEqual (Mat s a b) Source #
Instance details Defined in Algebra.Linear Methods (=.=) :: Mat s a b -> Mat s a b -> Property Source #
type Obj (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear type Obj (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) = VectorR

newtype Flat w v s Source #

View of the matrix as a composition of functors.

Constructors

Flat
Fields fromFlat :: w (v s)

Instances

Instances details

(Foldable w, Foldable v) => Foldable (Flat w v) Source #
Instance details Defined in Algebra.Linear Methods fold :: Monoid m => Flat w v m -> m # foldMap :: Monoid m => (a -> m) -> Flat w v a -> m # foldMap' :: Monoid m => (a -> m) -> Flat w v a -> m # foldr :: (a -> b -> b) -> b -> Flat w v a -> b # foldr' :: (a -> b -> b) -> b -> Flat w v a -> b # foldl :: (b -> a -> b) -> b -> Flat w v a -> b # foldl' :: (b -> a -> b) -> b -> Flat w v a -> b # foldr1 :: (a -> a -> a) -> Flat w v a -> a # foldl1 :: (a -> a -> a) -> Flat w v a -> a # toList :: Flat w v a -> [a] # null :: Flat w v a -> Bool # length :: Flat w v a -> Int # elem :: Eq a => a -> Flat w v a -> Bool # maximum :: Ord a => Flat w v a -> a # minimum :: Ord a => Flat w v a -> a # sum :: Num a => Flat w v a -> a # product :: Num a => Flat w v a -> a #
(Applicative w, Applicative v) => Applicative (Flat w v) Source #
Instance details Defined in Algebra.Linear Methods pure :: a -> Flat w v a # (<>) :: Flat w v (a -> b) -> Flat w v a -> Flat w v b # liftA2 :: (a -> b -> c) -> Flat w v a -> Flat w v b -> Flat w v c # (>) :: Flat w v a -> Flat w v b -> Flat w v b # (<*) :: Flat w v a -> Flat w v b -> Flat w v a #
(Functor w, Functor v) => Functor (Flat w v) Source #
Instance details Defined in Algebra.Linear Methods fmap :: (a -> b) -> Flat w v a -> Flat w v b # (<$) :: a -> Flat w v b -> Flat w v a #
Show (w (v s)) => Show (Flat w v s) Source #
Instance details Defined in Algebra.Linear Methods showsPrec :: Int -> Flat w v s -> ShowS # show :: Flat w v s -> String # showList :: [Flat w v s] -> ShowS #

flatMat :: Mat s w v -> Flat w v s Source #

matFlat :: Flat w v s -> Mat s w v Source #

fromRel :: (VectorR a, VectorR b) => Rel s (Rep a) (Rep b) -> Mat s a b Source #

type Mat3x3 s = SqMat V3 s Source #

type Mat2x2 s = SqMat V2 s Source #

pattern Mat2x2 :: forall s. s -> s -> s -> s -> Mat s V2 V2 Source #

pattern Mat3x3 :: forall s. s -> s -> s -> s -> s -> s -> s -> s -> s -> Mat s V3 V3 Source #

rotation2d :: Transcendental a => a -> Mat2x2 a Source #

crossProductMatrix :: Group a => V3 a -> Mat3x3 a Source #

outerWith :: (Applicative v, Applicative w) => (s -> t -> u) -> w s -> v t -> Mat u v w Source #

outer :: (Applicative v, Applicative w, Multiplicative s) => Euclid w s -> Euclid v s -> Mat s (Euclid w) (Euclid v) Source #

Outer product

diagonal :: Eq (Rep v) => Representable v => Ring s => Applicative v => v s -> SqMat v s Source #

rotation3d :: Transcendental a => a -> V3 a -> Mat3x3 a Source #

3d rotation around given axis

rotationFromTo :: forall a. Algebraic a => V3 a -> V3 a -> Mat3x3 a Source #

3d rotation mapping the direction of from to that of to

transpose :: Functor f => Distributive g => Mat a f g -> Mat a g f Source #

Transposition as distribution

matMul :: (Foldable u, Ring s, Applicative w, Applicative v, Applicative u) => Mat s u w -> Mat s v u -> Mat s v w Source #

(<+>) :: (Applicative f, Additive b) => f b -> f b -> f b Source #

matVecMul :: forall s v w. (Ring s, Foldable v, Applicative v, Applicative w) => Mat s v w -> v s -> w s Source #

prop_linear_with_functor_laws :: Property Source #

runTests :: IO Bool Source #