Safe Haskell	Safe
Language	Haskell2010

Math.Matrix.Matrix

Contents

Orphan instances

Synopsis

type family Transpose a
transposeIso :: (Linearizable LinearMap (:*:) m n a, Linearizable LinearMap (:*:) n m a, Transposable n m a, Transposable m n a) => (m a :-> n a) :==: (n a :-> m a)
vnegateIso :: VectorSpace v => v :==: v
(!$!) :: (Linearizable LinearMap (:*:) f g b, Num b, Functor f, Functor g) => f (a -> b) -> g a -> f b :-> g b
matrixBind :: Functor f => f a -> (a -> g b) -> (f :*: g) b
matrix2d :: (Functor f, Functor g) => (a -> b -> c) -> f a -> g b -> (f :*: g) c
matrix3d :: (Functor f, Functor g, Functor h) => (a -> b -> c -> d) -> f a -> g b -> h c -> (f :*: (g :*: h)) d
matrix4d :: (Functor f, Functor g, Functor h, Functor i) => (a -> b -> c -> d -> e) -> f a -> g b -> h c -> i d -> (f :*: (g :*: (h :*: i))) e
constantMatrix :: (Applicative m, Applicative n) => a -> (m :*: n) a
applyCol :: Applicative f => (f :*: (->) a) b -> f a -> f b
applyRow :: ((->) a :*: g) b -> a -> g b
matrixA :: (ArrowApply arr, FunctorArrow f arr arr, FunctorArrow g arr arr) => arr (a, b) c -> arr (f a, g b) ((f :*: g) c)
mod_linear_map :: (Num a, Linearizable LinearMap (:*:) f f a, Diagonalizable f a, Integral (f a)) => f a -> f a :-> f a
modularExponentiation :: (SupportsMatrixMultiplication f f f (g b), Scalar (f (g b)) ~ g b, Linearizable LinearMap (:*:) f f (g b), Linearizable LinearMap (:*:) g g b, Diagonalizable g b, LinearTransform f f b, FunctorArrow f LinearMap LinearMap, Integral b) => (f (g b) :-> f (g b)) -> Integer -> b -> f (g b) :-> f (g b)
isSymmetric :: (Eq (m (m a)), LinearTransform m m a, Linearizable LinearMap (:*:) m m a, Diagonalizable m a, Transposable m m a, Applicative m, Foldable m, Eq a) => (m a :-> m a) -> Bool
applyVector :: (Num b, LinearTransform m m b, Linearizable LinearMap (:*:) m m b, Transposable m m b, Diagonalizable m b) => m (a -> b) -> m a -> m b
applyColumns :: (Functor n, Functor m, Applicative f) => (m :*: f) (a -> b) -> (n :*: f) a -> (m :*: n) (f b)
matrixAssoc :: Functor f => ((f :*: g) :*: h) a -> (f :*: (g :*: h)) a
matrixUnassoc :: Functor f => (f :*: (g :*: h)) a -> ((f :*: g) :*: h) a
matrixLeftId :: (I :*: f) a -> f a
matrixRightId :: Functor f => (f :*: I) a -> f a
(!*!) :: (Functor f, Functor g, Num a) => g a -> f a -> (g :*: f) a
(<!!>) :: (Ix i, Ix j) => (Array i :*: Array j) a -> (i, j) -> a
projectDown :: (Functor (m \\\ m'), ProjectionSpace m m', ProjectionSpace n Vector1) => (m :*: n) a -> (m \\\ m') a
projectRight :: (ProjectionSpace m Vector1, ProjectionSpace n n') => (m :*: n) a -> (n \\\ n') a
projectDiagonal :: (ProjectionSpace m m', ProjectionSpace n n') => (m :*: n) a -> (m' :*: (n \\\ n')) a
matrixCommutator :: (VectorSpace ((f :*: f) a), InnerProductSpaceFunctor f a, ConjugateSymmetric a, Transposable f f a, Num a) => (f :*: f) a -> (f :*: f) a -> (f :*: f) a
matrixAnticommutator :: forall {g :: Type -> Type} {a}. (Scalar (g a) ~ a, InnerProductSpaceFunctor g a, ConjugateSymmetric a, Transposable g g a, VectorSpace ((g :*: g) a), Num a) => (g :*: g) a -> (g :*: g) a -> (g :*: g) a
normalizedAnticommutator :: forall {g :: Type -> Type} {a}. (Scalar (g a) ~ a, Fractional (Scalar ((g :*: g) a)), InnerProductSpaceFunctor g a, ConjugateSymmetric a, Transposable g g a, Num a, VectorSpace ((g :*: g) a)) => (g :*: g) a -> (g :*: g) a -> (g :*: g) a
commute :: forall {f :: Type -> Type} {b}. (Scalar (f b) ~ b, Fractional (Scalar ((f :*: f) b)), InnerProductSpaceFunctor f b, ConjugateSymmetric b, Transposable f f b, VectorSpace ((f :*: f) b), Num b) => Complex ((f :*: f) b) -> Complex ((f :*: f) b) -> ((f :*: f) :*: (f :*: f)) (Complex b)
bind_diagonal :: Monad m => (m :*: m) a -> (a -> m b) -> m b
pairMatrix :: (Functor f, Functor g) => f a -> g b -> (f :*: g) (a, b)
(!+!) :: (Functor f, Functor g, Num a) => g a -> f a -> (g :*: f) a
cofunctorInverseImage :: (Functor f, CoFunctor g) => (a -> b) -> (g :*: f) b -> (g :*: f) a
cofunctorMap :: (CoFunctor f, CoFunctor g) => (a -> b) -> (g :*: f) a -> (g :*: f) b
inTranspose :: (Monad g, Indexable f b, Integral b) => (f :*: g) a -> (a -> (g :*: f) b) -> (f :*: g) b
fmap_split :: (SplittableVector m n, AppendableVector m' n') => (m a -> m' a') -> (n a -> n' a') -> (m :+: n) a -> (m' :+: n') a'
split_matrix :: (SplittableVector m n, SplittableVector f f', Functor (f :+: f'), Functor (m :+: n)) => ((f :+: f') :*: (m :+: n)) a -> (((f :*: m) a, (f :*: n) a), ((f' :*: m) a, (f' :*: n) a))
append_matrix :: (AppendableVector m' n', AppendableVector m n) => (m :*: m') a -> (m :*: n') a -> (n :*: m') a -> (n :*: n') a -> ((m :+: n) :*: (m' :+: n')) a
matrix_iso :: f (g a) :==: (f :*: g) a
matrix_size :: (CoordinateSpace (g (f a)), CoordinateSpace (f (g a)), Transposable f g a) => (f :*: g) a -> (Int, Int)
data Strassen_Split f g a where
- Strassen_Two :: (Vector2 :*: Vector2) a -> Strassen_Split Vector1 Vector1 a
- Strassen_PowerOfTwo :: Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a
strassen_plus :: Num a => Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a
strassen_minus :: Num a => Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a
strassen_matrix_multiply :: (ConjugateSymmetric a, Num a) => Strassen_Split f g a -> Strassen_Split g h a -> Strassen_Split f h a

Documentation

type family Transpose a Source #

Instances

Instances details

type Transpose ((f :*: g) a) Source #
Instance details Defined in Math.Matrix.Matrix type Transpose ((f :: g) a) = (g :: f) a

transposeIso :: (Linearizable LinearMap (:*:) m n a, Linearizable LinearMap (:*:) n m a, Transposable n m a, Transposable m n a) => (m a :-> n a) :==: (n a :-> m a) Source #

vnegateIso :: VectorSpace v => v :==: v Source #

(!$!) :: (Linearizable LinearMap (:*:) f g b, Num b, Functor f, Functor g) => f (a -> b) -> g a -> f b :-> g b Source #

matrixBind :: Functor f => f a -> (a -> g b) -> (f :*: g) b Source #

matrix2d :: (Functor f, Functor g) => (a -> b -> c) -> f a -> g b -> (f :*: g) c Source #

this is same as Matrix.Interface.matrix

matrix3d :: (Functor f, Functor g, Functor h) => (a -> b -> c -> d) -> f a -> g b -> h c -> (f :*: (g :*: h)) d Source #

matrix4d :: (Functor f, Functor g, Functor h, Functor i) => (a -> b -> c -> d -> e) -> f a -> g b -> h c -> i d -> (f :*: (g :*: (h :*: i))) e Source #

constantMatrix :: (Applicative m, Applicative n) => a -> (m :*: n) a Source #

needs type annotation

applyCol :: Applicative f => (f :*: (->) a) b -> f a -> f b Source #

applyRow :: ((->) a :*: g) b -> a -> g b Source #

matrixA :: (ArrowApply arr, FunctorArrow f arr arr, FunctorArrow g arr arr) => arr (a, b) c -> arr (f a, g b) ((f :*: g) c) Source #

mod_linear_map :: (Num a, Linearizable LinearMap (:*:) f f a, Diagonalizable f a, Integral (f a)) => f a -> f a :-> f a Source #

https://en.wikipedia.org/wiki/Modular_exponentiation modular_exponentiation m b c == m^b (mod c)

modularExponentiation :: (SupportsMatrixMultiplication f f f (g b), Scalar (f (g b)) ~ g b, Linearizable LinearMap (:*:) f f (g b), Linearizable LinearMap (:*:) g g b, Diagonalizable g b, LinearTransform f f b, FunctorArrow f LinearMap LinearMap, Integral b) => (f (g b) :-> f (g b)) -> Integer -> b -> f (g b) :-> f (g b) Source #

isSymmetric :: (Eq (m (m a)), LinearTransform m m a, Linearizable LinearMap (:*:) m m a, Diagonalizable m a, Transposable m m a, Applicative m, Foldable m, Eq a) => (m a :-> m a) -> Bool Source #

applyVector :: (Num b, LinearTransform m m b, Linearizable LinearMap (:*:) m m b, Transposable m m b, Diagonalizable m b) => m (a -> b) -> m a -> m b Source #

applyColumns :: (Functor n, Functor m, Applicative f) => (m :*: f) (a -> b) -> (n :*: f) a -> (m :*: n) (f b) Source #

matrixAssoc :: Functor f => ((f :*: g) :*: h) a -> (f :*: (g :*: h)) a Source #

matrixUnassoc :: Functor f => (f :*: (g :*: h)) a -> ((f :*: g) :*: h) a Source #

matrixLeftId :: (I :*: f) a -> f a Source #

matrixRightId :: Functor f => (f :*: I) a -> f a Source #

(!*!) :: (Functor f, Functor g, Num a) => g a -> f a -> (g :*: f) a Source #

(<!!>) :: (Ix i, Ix j) => (Array i :*: Array j) a -> (i, j) -> a Source #

projectDown :: (Functor (m \\\ m'), ProjectionSpace m m', ProjectionSpace n Vector1) => (m :*: n) a -> (m \\\ m') a Source #

projectRight :: (ProjectionSpace m Vector1, ProjectionSpace n n') => (m :*: n) a -> (n \\\ n') a Source #

projectDiagonal :: (ProjectionSpace m m', ProjectionSpace n n') => (m :*: n) a -> (m' :*: (n \\\ n')) a Source #

matrixCommutator :: (VectorSpace ((f :*: f) a), InnerProductSpaceFunctor f a, ConjugateSymmetric a, Transposable f f a, Num a) => (f :*: f) a -> (f :*: f) a -> (f :*: f) a Source #

matrixAnticommutator :: forall {g :: Type -> Type} {a}. (Scalar (g a) ~ a, InnerProductSpaceFunctor g a, ConjugateSymmetric a, Transposable g g a, VectorSpace ((g :*: g) a), Num a) => (g :*: g) a -> (g :*: g) a -> (g :*: g) a Source #

normalizedAnticommutator :: forall {g :: Type -> Type} {a}. (Scalar (g a) ~ a, Fractional (Scalar ((g :*: g) a)), InnerProductSpaceFunctor g a, ConjugateSymmetric a, Transposable g g a, Num a, VectorSpace ((g :*: g) a)) => (g :*: g) a -> (g :*: g) a -> (g :*: g) a Source #

commute :: forall {f :: Type -> Type} {b}. (Scalar (f b) ~ b, Fractional (Scalar ((f :*: f) b)), InnerProductSpaceFunctor f b, ConjugateSymmetric b, Transposable f f b, VectorSpace ((f :*: f) b), Num b) => Complex ((f :*: f) b) -> Complex ((f :*: f) b) -> ((f :*: f) :*: (f :*: f)) (Complex b) Source #

bind_diagonal :: Monad m => (m :*: m) a -> (a -> m b) -> m b Source #

pairMatrix :: (Functor f, Functor g) => f a -> g b -> (f :*: g) (a, b) Source #

(!+!) :: (Functor f, Functor g, Num a) => g a -> f a -> (g :*: f) a Source #

cofunctorInverseImage :: (Functor f, CoFunctor g) => (a -> b) -> (g :*: f) b -> (g :*: f) a Source #

cofunctorMap :: (CoFunctor f, CoFunctor g) => (a -> b) -> (g :*: f) a -> (g :*: f) b Source #

inTranspose :: (Monad g, Indexable f b, Integral b) => (f :*: g) a -> (a -> (g :*: f) b) -> (f :*: g) b Source #

fmap_split :: (SplittableVector m n, AppendableVector m' n') => (m a -> m' a') -> (n a -> n' a') -> (m :+: n) a -> (m' :+: n') a' Source #

split_matrix :: (SplittableVector m n, SplittableVector f f', Functor (f :+: f'), Functor (m :+: n)) => ((f :+: f') :*: (m :+: n)) a -> (((f :*: m) a, (f :*: n) a), ((f' :*: m) a, (f' :*: n) a)) Source #

append_matrix :: (AppendableVector m' n', AppendableVector m n) => (m :*: m') a -> (m :*: n') a -> (n :*: m') a -> (n :*: n') a -> ((m :+: n) :*: (m' :+: n')) a Source #

matrix_iso :: f (g a) :==: (f :*: g) a Source #

matrix_size :: (CoordinateSpace (g (f a)), CoordinateSpace (f (g a)), Transposable f g a) => (f :*: g) a -> (Int, Int) Source #

data Strassen_Split f g a where Source #

Constructors

Strassen_Two :: (Vector2 :*: Vector2) a -> Strassen_Split Vector1 Vector1 a
Strassen_PowerOfTwo :: Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a

strassen_plus :: Num a => Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a Source #

strassen_minus :: Num a => Strassen_Split f g a -> Strassen_Split f g a -> Strassen_Split f g a Source #

strassen_matrix_multiply :: (ConjugateSymmetric a, Num a) => Strassen_Split f g a -> Strassen_Split g h a -> Strassen_Split f h a Source #

https://en.wikipedia.org/wiki/Strassen_algorithm

Orphan instances

Ix i => PpShowF (Array i) Source #
Instance details Methods ppf :: PpShow a => Array i a -> Doc Source #
Ix i => PpShowVerticalF (Array i) Source #
Instance details Methods ppfVertical :: PpShow a => Array i a -> Doc Source #
(MonadZip g, MonadZip f, forall b. Transposable f g b) => MonadZip (g :*: f) Source #
Instance details Methods mzip :: (g :: f) a -> (g :: f) b -> (g :: f) (a, b) # mzipWith :: (a -> b -> c) -> (g :: f) a -> (g :: f) b -> (g :: f) c # munzip :: (g :: f) (a, b) -> ((g :: f) a, (g :*: f) b) #
(Contravariant f, Functor g) => Contravariant (g :*: f) Source #
Instance details Methods contramap :: (a' -> a) -> (g :: f) a -> (g :: f) a' # (>$) :: b -> (g :: f) b -> (g :: f) a #
(MonadPlus g, MonadPlus f, forall b. Transposable f g b) => MonadPlus (g :*: f) Source #
Instance details Methods mzero :: (g :: f) a # mplus :: (g :: f) a -> (g :: f) a -> (g :: f) a #
(Comonad f, Comonad g, forall a. Transposable g f a, forall b. Transposable f g b, forall a. Diagonalizable f (f (g a)), forall a. LinearTransform g f (f (g a)), forall a. Diagonalizable g (g a), forall a. LinearTransform f g (g a)) => Comonad (f :*: g) Source #	See video by Bartosz Milewski ("Category theory II 7.2: Comonads categorically and examples")
Instance details Methods extract :: (f :: g) a -> a Source # duplicate :: (f :: g) a -> (f :: g) ((f :: g) a) Source # extend :: ((f :: g) a -> b) -> (f :: g) a -> (f :: g) b Source # (=>>) :: (f :: g) a -> ((f :: g) a -> b) -> (f :: g) b Source # (.>>) :: (f :: g) a -> b -> (f :: g) b Source #
(Num a, ConjugateSymmetric a) => LieAlgebra ((Vector3 :*: Vector3) a) Source #
Instance details Methods (%<>%) :: (Vector3 :: Vector3) a -> (Vector3 :: Vector3) a -> (Vector3 :*: Vector3) a Source #
(Num a, ConjugateSymmetric a) => LieAlgebra ((Vector4 :*: Vector4) a) Source #
Instance details Methods (%<>%) :: (Vector4 :: Vector4) a -> (Vector4 :: Vector4) a -> (Vector4 :*: Vector4) a Source #
(Functor f, Functor g) => Builder ((g :*: f) a) Source #
Instance details Associated Types data Unfold ((g :: f) a) :: Type -> Type Source # Methods build :: Unfold ((g :: f) a) a0 -> a0 -> (g :*: f) a Source #
(Eq a, Applicative f, Applicative g, Foldable f, Foldable g, Ord (f (g a))) => Ord ((f :*: g) a) Source #
Instance details Methods compare :: (f :: g) a -> (f :: g) a -> Ordering # (<) :: (f :: g) a -> (f :: g) a -> Bool # (<=) :: (f :: g) a -> (f :: g) a -> Bool # (>) :: (f :: g) a -> (f :: g) a -> Bool # (>=) :: (f :: g) a -> (f :: g) a -> Bool # max :: (f :: g) a -> (f :: g) a -> (f :: g) a # min :: (f :: g) a -> (f :: g) a -> (f :: g) a #