Safe Haskell	None
Language	Haskell2010

Algebra.Linear

Synopsis

type VectorSpace scalar a = (Field scalar, Module scalar a, Group a)
type VectorR v = (Applicative v, Traversable v)
class VectorR v => InnerProdSpace v where
- inner :: Field s => v s -> v s -> s
data VZero a = VZero
data VNext v a = VNext !(v a) !a
type V1' = VNext VZero
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 :: Field s => InnerProdSpace v => Floating s => v s -> s
normalize :: VectorSpace s (v s) => Floating s => InnerProdSpace v => v s -> v s
(×) :: Ring a => V3 a -> V3 a -> V3 a
index :: Applicative v => Traversable v => v Int
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
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
(*<) :: (Functor f, Multiplicative b) => b -> f b -> f b
(<+>) :: (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
rotation2d :: (Group a, Floating a) => a -> Mat2x2 a
crossProductMatrix :: Group a => V3 a -> Mat3x3 a
(⊗) :: (Applicative v, Applicative w, Multiplicative s) => w s -> v s -> Mat s w v
tensorWith :: (Applicative v, Applicative w) => (s -> t -> u) -> w s -> v t -> Mat u v w
identity :: Traversable v => Ring s => Applicative v => SqMat v s
diagonal :: Traversable v => Ring s => Applicative v => v s -> SqMat v s
rotation3d :: Ring a => Floating a => a -> V3 a -> Mat3x3 a
rotationFromTo :: (Floating a, Module a a, Field a) => V3 a -> V3 a -> Mat3x3 a
transpose :: Applicative g => Traversable f => Mat a f g -> Mat a g f
matMul :: (Traversable u, Ring s, Applicative w, Applicative v, Applicative u) => Mat s u w -> Mat s v u -> Mat s v w
newtype OrthoMat v s = OrthoMat (SqMat v s)

Documentation

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

type VectorR v = (Applicative v, Traversable v) Source #

Representation of vector as traversable functor

class VectorR v => InnerProdSpace v where Source #

Methods

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

Instances

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

data VZero a Source #

Constructors

VZero

Instances

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

data VNext v a Source #

Constructors

VNext !(v a) !a

Instances

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 #
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 #
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 # 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) #
(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 #
(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 #

type V1' = VNext VZero 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

(Applicative f, Module s a) => Module s (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (*^) :: s -> Euclid f a -> Euclid f a Source #
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 #
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 #
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 # 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) #
VectorR f => InnerProdSpace (Euclid f) Source #
Instance details Defined in Algebra.Linear Methods inner :: Field s => Euclid f s -> Euclid f s -> s 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 #
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, Group a) => Group (Euclid f a) Source #
Instance details Defined in Algebra.Linear Methods (-) :: 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 #
(Applicative f, AbelianAdditive a) => AbelianAdditive (Euclid f a) Source #
Instance details Defined in Algebra.Linear
(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 #

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

normalize :: VectorSpace s (v s) => Floating 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

index :: Applicative v => Traversable v => v Int Source #

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

(Applicative f, Applicative g, Module s a) => Module 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 => Category (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear Associated Types type Con a :: Constraint Source # Methods (.) :: (Con a, Con b, Con c) => Mat s b c -> Mat s a b -> Mat s a c Source # id :: Con a => Mat s a a Source #
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, 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 # negate :: Mat a f g -> Mat a f g Source # mult :: Integer -> Mat a f g -> Mat a f g Source #
(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 #

newtype Flat w v s Source #

View of the matrix as a composition of functors.

Constructors

Flat
Fields fromFlat :: w (v s)

Instances

(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 #
(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 #
(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 # 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 #
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 #

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 #

(*<) :: (Functor f, Multiplicative b) => b -> f b -> f b infixr 7 Source #

Vector scaling. If Module a (f a), then (*^) must be the same as (*<).

(<+>) :: (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 #

rotation2d :: (Group a, Floating a) => a -> Mat2x2 a Source #

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

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

Tensor product

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

identity :: Traversable v => Ring s => Applicative v => SqMat v s Source #

diagonal :: Traversable v => Ring s => Applicative v => v s -> SqMat v s Source #

rotation3d :: Ring a => Floating a => a -> V3 a -> Mat3x3 a Source #

3d rotation around given axis

rotationFromTo :: (Floating a, Module a a, Field a) => V3 a -> V3 a -> Mat3x3 a Source #

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

transpose :: Applicative g => Traversable f => Mat a f g -> Mat a g f Source #

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

newtype OrthoMat v s Source #

Constructors

OrthoMat (SqMat v s)

Instances

(Ring s, Applicative v, Traversable v) => Division (OrthoMat v s) Source #
Instance details Defined in Algebra.Linear Methods recip :: OrthoMat v s -> OrthoMat v s Source # (/) :: OrthoMat v s -> OrthoMat v s -> OrthoMat v s Source # (^) :: OrthoMat v s -> Integer -> OrthoMat v s Source #
(Ring s, Applicative v, Traversable v) => Multiplicative (OrthoMat v s) Source #
Instance details Defined in Algebra.Linear Methods (*) :: OrthoMat v s -> OrthoMat v s -> OrthoMat v s Source # one :: OrthoMat v s Source # (^+) :: OrthoMat v s -> Natural -> OrthoMat v s Source #