Safe Haskell	Safe
Language	Haskell2010

Math.Matrix.Simple

Contents

Orphan instances

Description

This module provides matrices with indices. Matrices are constructed using smatrix and svec operations, example:

let m = smatrix (*) (svec3 1 2 3) (svec4 5 4 3 2) :: (ThreeD :&: FourD) Int
== 5   4 3 2
   10  8 6 4
   15 12 9 6

indexing occurs with the (#) operator, for example, m # (1,0) == 10.

The number after svec is the number of elements in the dimension along row or column. Indices start from zero. The binary operation is applied to each pair of elements to construct the matrix element.

Matrices can also be constructed using Matrix constructor, with indices as input, e.g.:

identity :: (ThreeD :&: ThreeD) Integer

identity = Matrix $ \i j -> if i == j then 1 else 0

Note that index types are specifically restricted to be small types, since the Universe type class operations are often used to enumerate all elements of the type. This is enforced in show operations, which only show ten elements per dimension. For convenience, pre-defined index types, OneD, TwoD, ThreeD, FourD, FiveD are defined, for up to 5 dimensional matrices.

Synopsis

type (:&:) row col elem = (row -> elem) :-> (col -> elem)
type Matrix3S row col dep elem = (row -> dep -> elem) :-> (col :&: dep) elem
type Matrix4S row col dep tim elem = (row -> (dep :&: tim) elem) :-> Matrix3S col dep tim elem
type MatrixA arr row col elem = arr row elem :-> arr col elem
type SMatrix1 elem = (OneD :&: OneD) elem
type SMatrix2 elem = (TwoD :&: TwoD) elem
type SMatrix3 elem = (ThreeD :&: ThreeD) elem
type SMatrix4 elem = (FourD :&: FourD) elem
type InfMatrix elem = (Integer :&: Integer) elem
type RatioMatrix elem = (Rational :&: Rational) elem
mapDimensions :: (Universe a, Universe b, Integral a, ConjugateSymmetric c, Num c) => ((->) a :~> f) -> ((->) b :~> g) -> (a :&: b) c -> (f :*: g) c
simpleMatrix :: (Integral a, Universe b, Universe a, ConjugateSymmetric c, Num c, Functor m, Functor n) => (a :&: b) c -> m a -> n b -> (m :*: n) c
index_delta :: (Eq ind, Num a) => ind -> a -> (ind -> a) -> ind -> a
partial_derivate_ind :: (Integral ind, Universe ind, Eq ind, ConjugateSymmetric a, Closed a, Infinitesimal Stream a) => ind -> Dual (ind -> a) -> Dual (ind -> a)
partial_derivate_list :: (Universe a, Eq a, Integral a, ConjugateSymmetric b, Closed b, Infinitesimal Stream b) => [Dual (a -> b) -> Dual (a -> b)]
divergence_index :: (ConjugateSymmetric b, Closed b, Infinitesimal Stream b, Integral ind, Eq ind, Universe a, Universe ind) => ((ind -> b) :-> (a -> b)) -> Dual (ind -> b)
grad_index :: (Scalar (ind -> a) ~ a, Integral ind, Universe ind, ConjugateSymmetric a, Closed a, Infinitesimal Stream a, Eq ind) => Dual (ind -> a) -> (ind -> a) :-> (ind -> a)
laplace_index :: (ConjugateSymmetric a, Closed a, Infinitesimal Stream a, Integral ind, Eq ind, Universe ind) => Dual (ind -> a) -> Dual (ind -> a)
set_index :: Eq ind => ind -> s -> (ind -> s) -> ind -> s
update_row_index :: Eq ind => g a -> ind -> ((->) ind :*: g) a -> ((->) ind :*: g) a
update_column_index :: (Applicative f, Eq ind) => f a -> ind -> (f :*: (->) ind) a -> (f :*: (->) ind) a
sumS :: (Num b, Universe a) => (a -> b) -> b
productS :: (Num b, Universe a) => (a -> b) -> b
cov_index :: (Integral a, Universe a, Num (Scalar a), ConjugateSymmetric (Scalar a)) => a -> Dual (a -> Scalar a)
kronecker_delta :: (Eq a, Num b) => a -> a -> b
(%***%) :: (Integral row, Integral mid, Universe row, Universe col, Universe mid, SupportsMatrixMultiplication ((->) row) ((->) col) ((->) mid) a) => (row :&: mid) a -> (mid :&: col) a -> (row :&: col) a
class CoFactorDimension dim where
- cofactorDim :: (Num a, ConjugateSymmetric a) => dim -> (dim :&: dim) a -> a
leviCivita2S :: Num a => ((->) TwoD :*: (->) TwoD) a
leviCivita3S :: ((FUN 'Many ThreeD :: Type -> Type) :*: ((FUN 'Many ThreeD :: Type -> Type) :*: (FUN 'Many ThreeD :: Type -> Type))) Integer
leviCivita4S :: ((FUN 'Many FourD :: Type -> Type) :*: ((FUN 'Many FourD :: Type -> Type) :*: ((FUN 'Many FourD :: Type -> Type) :*: (FUN 'Many FourD :: Type -> Type)))) Integer
leviCivitaS :: (Num b, Enum a, Universe a) => (a -> b) -> b
leviCivita :: Num a => [a] -> a
crossProductS :: (Integral a, Num b, Universe a) => (a -> b) -> (a -> b) -> a -> b
crossProduct :: (Integral a, Monoid a) => [a] -> [a] -> [a]
smatrix :: (Universe col, Integral row, Universe row, ConjugateSymmetric c, Num c) => (a -> b -> c) -> (row -> a) -> (col -> b) -> (row :&: col) c
swapRows :: (Num a, ConjugateSymmetric a, Eq row, Integral row, Universe row, Universe col) => row -> row -> (row :&: col) a -> (row :&: col) a
mulRow :: (Eq row, ConjugateSymmetric a, Num a, Integral row, Universe row, Universe col) => a -> row -> (row :&: col) a -> (row :&: col) a
swapIndex :: Eq a => a -> a -> a -> a
smatrixA :: (Arrow arr, Diagonalizable (arr row) c, Linearizable LinearMap (:*:) (arr row) (arr col) c, LinearTransform (arr row) (arr col) c) => arr (a, b) c -> arr row a -> arr col b -> MatrixA arr row col c
smatrix3 :: (a -> b -> c -> d) -> (row -> a) -> (col -> b) -> (depth -> c) -> ((->) row :*: ((->) col :*: (->) depth)) d
function_smatrix :: (Universe a, Universe b, Integral a, ConjugateSymmetric c, Num c) => (a -> b -> c) -> (a :&: b) c
smatrix_inverse_image :: (ConjugateSymmetric a, Num a, Integral row, Integral row', Universe col', Universe row', Universe row, Universe col) => ((row, col) -> (row', col')) -> (row' :&: col') a -> (row :&: col) a
inverse_image_smatrix :: (Integral row', Integral row, Universe row, ConjugateSymmetric a, Num a, Universe row', Universe col', Universe col) => (row -> row', col -> col') -> (row' :&: col') a -> (row :&: col) a
sapply :: (Integral (a -> b), ConjugateSymmetric b, Num b, Eq a, Universe a, Universe b) => ((a -> b) :&: a) b
(#) :: (ConjugateSymmetric a, Integral row, Num a, Universe row, Universe col) => (row :&: col) a -> (row, col) -> a
indexA :: (ArrowApply arr, Diagonalizable (arr row) a, Linearizable LinearMap (:*:) (arr row) (arr col) a, LinearTransform (arr row) (arr col) a) => MatrixA arr row col a -> arr (row, col) a
sdeterminant2 :: (ConjugateSymmetric a, Num a) => (TwoD :&: TwoD) a -> a
cofactor4_11 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a
cofactor4_21 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a
cofactor4_31 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a
cofactor4_41 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a
cofactor3_11 :: (ConjugateSymmetric a, Num a) => (ThreeD :&: ThreeD) a -> a
cofactor3_21 :: (ConjugateSymmetric a, Num a) => (ThreeD :&: ThreeD) a -> a
cofactor3_31 :: (ConjugateSymmetric a, Num a) => (ThreeD :&: ThreeD) a -> a
cofactor2_11 :: (ConjugateSymmetric a, Num a) => (TwoD :&: TwoD) a -> a
cofactor2_21 :: (ConjugateSymmetric a, Num a) => (TwoD :&: TwoD) a -> a
cofactorDim2 :: (ConjugateSymmetric a, Num a) => TwoD -> (TwoD :&: TwoD) a -> a
cofactorDim3 :: (ConjugateSymmetric a, Num a) => ThreeD -> (ThreeD :&: ThreeD) a -> a
cofactorDim4 :: (ConjugateSymmetric a, Num a) => FourD -> (FourD :&: FourD) a -> a
sdeterminant :: (ConjugateSymmetric a, Num a, Integral dim, Universe dim, CoFactorDimension dim) => (dim :&: dim) a -> a
sdeterminant3 :: (Num a, ConjugateSymmetric a) => (ThreeD :&: ThreeD) a -> a
sdeterminant4 :: (Num a, ConjugateSymmetric a) => (FourD :&: FourD) a -> a
svec_inf :: Stream a -> Integer -> a
srotate_x :: (Floating a, ConjugateSymmetric a) => a -> SMatrix3 a

Documentation

type (:&:) row col elem = (row -> elem) :-> (col -> elem) Source #

type Matrix3S row col dep elem = (row -> dep -> elem) :-> (col :&: dep) elem Source #

type Matrix4S row col dep tim elem = (row -> (dep :&: tim) elem) :-> Matrix3S col dep tim elem Source #

type MatrixA arr row col elem = arr row elem :-> arr col elem Source #

type SMatrix1 elem = (OneD :&: OneD) elem Source #

type SMatrix2 elem = (TwoD :&: TwoD) elem Source #

type SMatrix3 elem = (ThreeD :&: ThreeD) elem Source #

type SMatrix4 elem = (FourD :&: FourD) elem Source #

type InfMatrix elem = (Integer :&: Integer) elem Source #

type RatioMatrix elem = (Rational :&: Rational) elem Source #

mapDimensions :: (Universe a, Universe b, Integral a, ConjugateSymmetric c, Num c) => ((->) a :~> f) -> ((->) b :~> g) -> (a :&: b) c -> (f :*: g) c Source #

simpleMatrix :: (Integral a, Universe b, Universe a, ConjugateSymmetric c, Num c, Functor m, Functor n) => (a :&: b) c -> m a -> n b -> (m :*: n) c Source #

index_delta :: (Eq ind, Num a) => ind -> a -> (ind -> a) -> ind -> a Source #

partial_derivate_ind :: (Integral ind, Universe ind, Eq ind, ConjugateSymmetric a, Closed a, Infinitesimal Stream a) => ind -> Dual (ind -> a) -> Dual (ind -> a) Source #

partial_derivate_list :: (Universe a, Eq a, Integral a, ConjugateSymmetric b, Closed b, Infinitesimal Stream b) => [Dual (a -> b) -> Dual (a -> b)] Source #

divergence_index :: (ConjugateSymmetric b, Closed b, Infinitesimal Stream b, Integral ind, Eq ind, Universe a, Universe ind) => ((ind -> b) :-> (a -> b)) -> Dual (ind -> b) Source #

grad_index :: (Scalar (ind -> a) ~ a, Integral ind, Universe ind, ConjugateSymmetric a, Closed a, Infinitesimal Stream a, Eq ind) => Dual (ind -> a) -> (ind -> a) :-> (ind -> a) Source #

laplace_index :: (ConjugateSymmetric a, Closed a, Infinitesimal Stream a, Integral ind, Eq ind, Universe ind) => Dual (ind -> a) -> Dual (ind -> a) Source #

set_index :: Eq ind => ind -> s -> (ind -> s) -> ind -> s Source #

update_row_index :: Eq ind => g a -> ind -> ((->) ind :*: g) a -> ((->) ind :*: g) a Source #

update_column_index :: (Applicative f, Eq ind) => f a -> ind -> (f :*: (->) ind) a -> (f :*: (->) ind) a Source #

sumS :: (Num b, Universe a) => (a -> b) -> b Source #

productS :: (Num b, Universe a) => (a -> b) -> b Source #

cov_index :: (Integral a, Universe a, Num (Scalar a), ConjugateSymmetric (Scalar a)) => a -> Dual (a -> Scalar a) Source #

kronecker_delta :: (Eq a, Num b) => a -> a -> b Source #

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

(%***%) :: (Integral row, Integral mid, Universe row, Universe col, Universe mid, SupportsMatrixMultiplication ((->) row) ((->) col) ((->) mid) a) => (row :&: mid) a -> (mid :&: col) a -> (row :&: col) a Source #

class CoFactorDimension dim where Source #

Methods

cofactorDim :: (Num a, ConjugateSymmetric a) => dim -> (dim :&: dim) a -> a Source #

Instances

Instances details

CoFactorDimension FourD Source #
Instance details Defined in Math.Matrix.Simple Methods cofactorDim :: (Num a, ConjugateSymmetric a) => FourD -> (FourD :&: FourD) a -> a Source #
CoFactorDimension ThreeD Source #
Instance details Defined in Math.Matrix.Simple Methods cofactorDim :: (Num a, ConjugateSymmetric a) => ThreeD -> (ThreeD :&: ThreeD) a -> a Source #
CoFactorDimension TwoD Source #
Instance details Defined in Math.Matrix.Simple Methods cofactorDim :: (Num a, ConjugateSymmetric a) => TwoD -> (TwoD :&: TwoD) a -> a Source #

leviCivita2S :: Num a => ((->) TwoD :*: (->) TwoD) a Source #

https://en.wikipedia.org/wiki/Levi-Civita_symbol Note indices are from 0..(n-1).

leviCivita3S :: ((FUN 'Many ThreeD :: Type -> Type) :*: ((FUN 'Many ThreeD :: Type -> Type) :*: (FUN 'Many ThreeD :: Type -> Type))) Integer Source #

https://en.wikipedia.org/wiki/Levi-Civita_symbol Note indices are from 0..(n-1).

leviCivita4S :: ((FUN 'Many FourD :: Type -> Type) :*: ((FUN 'Many FourD :: Type -> Type) :*: ((FUN 'Many FourD :: Type -> Type) :*: (FUN 'Many FourD :: Type -> Type)))) Integer Source #

https://en.wikipedia.org/wiki/Levi-Civita_symbol Note indices are from 0..(n-1).

leviCivitaS :: (Num b, Enum a, Universe a) => (a -> b) -> b Source #

https://en.wikipedia.org/wiki/Levi-Civita_symbol

leviCivita :: Num a => [a] -> a Source #

https://en.wikipedia.org/wiki/Levi-Civita_symbol Note indices are from 0..(n-1).

crossProductS :: (Integral a, Num b, Universe a) => (a -> b) -> (a -> b) -> a -> b Source #

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

crossProduct :: (Integral a, Monoid a) => [a] -> [a] -> [a] Source #

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

smatrix :: (Universe col, Integral row, Universe row, ConjugateSymmetric c, Num c) => (a -> b -> c) -> (row -> a) -> (col -> b) -> (row :&: col) c Source #

swapRows :: (Num a, ConjugateSymmetric a, Eq row, Integral row, Universe row, Universe col) => row -> row -> (row :&: col) a -> (row :&: col) a Source #

mulRow :: (Eq row, ConjugateSymmetric a, Num a, Integral row, Universe row, Universe col) => a -> row -> (row :&: col) a -> (row :&: col) a Source #

swapIndex :: Eq a => a -> a -> a -> a Source #

smatrixA :: (Arrow arr, Diagonalizable (arr row) c, Linearizable LinearMap (:*:) (arr row) (arr col) c, LinearTransform (arr row) (arr col) c) => arr (a, b) c -> arr row a -> arr col b -> MatrixA arr row col c Source #

smatrix3 :: (a -> b -> c -> d) -> (row -> a) -> (col -> b) -> (depth -> c) -> ((->) row :*: ((->) col :*: (->) depth)) d Source #

function_smatrix :: (Universe a, Universe b, Integral a, ConjugateSymmetric c, Num c) => (a -> b -> c) -> (a :&: b) c Source #

smatrix_inverse_image :: (ConjugateSymmetric a, Num a, Integral row, Integral row', Universe col', Universe row', Universe row, Universe col) => ((row, col) -> (row', col')) -> (row' :&: col') a -> (row :&: col) a Source #

inverse_image_smatrix :: (Integral row', Integral row, Universe row, ConjugateSymmetric a, Num a, Universe row', Universe col', Universe col) => (row -> row', col -> col') -> (row' :&: col') a -> (row :&: col) a Source #

sapply :: (Integral (a -> b), ConjugateSymmetric b, Num b, Eq a, Universe a, Universe b) => ((a -> b) :&: a) b Source #

(#) :: (ConjugateSymmetric a, Integral row, Num a, Universe row, Universe col) => (row :&: col) a -> (row, col) -> a Source #

indexA :: (ArrowApply arr, Diagonalizable (arr row) a, Linearizable LinearMap (:*:) (arr row) (arr col) a, LinearTransform (arr row) (arr col) a) => MatrixA arr row col a -> arr (row, col) a Source #

sdeterminant2 :: (ConjugateSymmetric a, Num a) => (TwoD :&: TwoD) a -> a Source #

cofactor4_11 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a Source #

cofactor4_21 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a Source #

cofactor4_31 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a Source #

cofactor4_41 :: (ConjugateSymmetric a, Num a) => (FourD :&: FourD) a -> a Source #

cofactor3_11 :: (ConjugateSymmetric a, Num a) => (ThreeD :&: ThreeD) a -> a Source #

cofactor3_21 :: (ConjugateSymmetric a, Num a) => (ThreeD :&: ThreeD) a -> a Source #

cofactor3_31 :: (ConjugateSymmetric a, Num a) => (ThreeD :&: ThreeD) a -> a Source #

cofactor2_11 :: (ConjugateSymmetric a, Num a) => (TwoD :&: TwoD) a -> a Source #

cofactor2_21 :: (ConjugateSymmetric a, Num a) => (TwoD :&: TwoD) a -> a Source #

cofactorDim2 :: (ConjugateSymmetric a, Num a) => TwoD -> (TwoD :&: TwoD) a -> a Source #

cofactorDim3 :: (ConjugateSymmetric a, Num a) => ThreeD -> (ThreeD :&: ThreeD) a -> a Source #

cofactorDim4 :: (ConjugateSymmetric a, Num a) => FourD -> (FourD :&: FourD) a -> a Source #

sdeterminant :: (ConjugateSymmetric a, Num a, Integral dim, Universe dim, CoFactorDimension dim) => (dim :&: dim) a -> a Source #

sdeterminant3 :: (Num a, ConjugateSymmetric a) => (ThreeD :&: ThreeD) a -> a Source #

sdeterminant4 :: (Num a, ConjugateSymmetric a) => (FourD :&: FourD) a -> a Source #

svec_inf :: Stream a -> Integer -> a Source #

srotate_x :: (Floating a, ConjugateSymmetric a) => a -> SMatrix3 a Source #

Orphan instances

(Num a, Universe col) => LinearTransform Vector1 ((->) col) a Source #
Instance details Methods (<>>) :: (col -> a) -> (Vector1 :: (->) col) a -> Vector1 a Source # (<<>) :: (Vector1 :: (->) col) a -> Vector1 a -> col -> a Source #
(Num a, Universe col) => LinearTransform Vector2 ((->) col) a Source #
Instance details Methods (<>>) :: (col -> a) -> (Vector2 :: (->) col) a -> Vector2 a Source # (<<>) :: (Vector2 :: (->) col) a -> Vector2 a -> col -> a Source #
(Num a, Universe col) => LinearTransform Vector3 ((->) col) a Source #
Instance details Methods (<>>) :: (col -> a) -> (Vector3 :: (->) col) a -> Vector3 a Source # (<<>) :: (Vector3 :: (->) col) a -> Vector3 a -> col -> a Source #
(Universe col, Num a) => LinearTransform Vector4 ((->) col) a Source #
Instance details Methods (<>>) :: (col -> a) -> (Vector4 :: (->) col) a -> Vector4 a Source # (<<>) :: (Vector4 :: (->) col) a -> Vector4 a -> col -> a Source #
Num a => DecomposableVectorSpace (Vector1 a) ((->) OneD) Source #
Instance details Methods decompose :: (Scalar (Vector1 a) -> res) -> Vector1 a -> OneD -> res Source # project :: Vector1 a -> OneD -> Scalar (Vector1 a) Source #
Num a => DecomposableVectorSpace (Vector2 a) ((->) TwoD) Source #
Instance details Methods decompose :: (Scalar (Vector2 a) -> res) -> Vector2 a -> TwoD -> res Source # project :: Vector2 a -> TwoD -> Scalar (Vector2 a) Source #
Num a => DecomposableVectorSpace (Vector3 a) ((->) ThreeD) Source #
Instance details Methods decompose :: (Scalar (Vector3 a) -> res) -> Vector3 a -> ThreeD -> res Source # project :: Vector3 a -> ThreeD -> Scalar (Vector3 a) Source #
Num a => DecomposableVectorSpace (Vector4 a) ((->) FourD) Source #
Instance details Methods decompose :: (Scalar (Vector4 a) -> res) -> Vector4 a -> FourD -> res Source # project :: Vector4 a -> FourD -> Scalar (Vector4 a) Source #
Num a => DecomposableVectorSpace (Stream a) ((->) Integer) Source #
Instance details Methods decompose :: (Scalar (Stream a) -> res) -> Stream a -> Integer -> res Source # project :: Stream a -> Integer -> Scalar (Stream a) Source #
Num a => Num (FourD -> a) Source #
Instance details Methods (+) :: (FourD -> a) -> (FourD -> a) -> FourD -> a # (-) :: (FourD -> a) -> (FourD -> a) -> FourD -> a # (*) :: (FourD -> a) -> (FourD -> a) -> FourD -> a # negate :: (FourD -> a) -> FourD -> a # abs :: (FourD -> a) -> FourD -> a # signum :: (FourD -> a) -> FourD -> a # fromInteger :: Integer -> FourD -> a #
Num a => Num (OneD -> a) Source #
Instance details Methods (+) :: (OneD -> a) -> (OneD -> a) -> OneD -> a # (-) :: (OneD -> a) -> (OneD -> a) -> OneD -> a # (*) :: (OneD -> a) -> (OneD -> a) -> OneD -> a # negate :: (OneD -> a) -> OneD -> a # abs :: (OneD -> a) -> OneD -> a # signum :: (OneD -> a) -> OneD -> a # fromInteger :: Integer -> OneD -> a #
Num a => Num (ThreeD -> a) Source #
Instance details Methods (+) :: (ThreeD -> a) -> (ThreeD -> a) -> ThreeD -> a # (-) :: (ThreeD -> a) -> (ThreeD -> a) -> ThreeD -> a # (*) :: (ThreeD -> a) -> (ThreeD -> a) -> ThreeD -> a # negate :: (ThreeD -> a) -> ThreeD -> a # abs :: (ThreeD -> a) -> ThreeD -> a # signum :: (ThreeD -> a) -> ThreeD -> a # fromInteger :: Integer -> ThreeD -> a #
Num a => Num (TwoD -> a) Source #
Instance details Methods (+) :: (TwoD -> a) -> (TwoD -> a) -> TwoD -> a # (-) :: (TwoD -> a) -> (TwoD -> a) -> TwoD -> a # (*) :: (TwoD -> a) -> (TwoD -> a) -> TwoD -> a # negate :: (TwoD -> a) -> TwoD -> a # abs :: (TwoD -> a) -> TwoD -> a # signum :: (TwoD -> a) -> TwoD -> a # fromInteger :: Integer -> TwoD -> a #
(Floating b, Universe a) => NormedSpace (a -> b) Source #
Instance details Methods norm :: (a -> b) -> Scalar (a -> b) Source # norm_squared :: (a -> b) -> Scalar (a -> b) Source #
(Integral a, Universe a, Num b, ConjugateSymmetric b) => Dualizable (a -> b) Dual Source #
Instance details Methods covector :: ((a -> b) -> Scalar (a -> b)) -> Dual (a -> b) Source # bracket :: Dual (a -> b) -> (a -> b) -> Scalar (a -> b) Source #
(b ~ Scalar a, Scalar (a -> b) ~ b, Integral a, VectorSpace b, ConjugateSymmetric b, Closed b, Infinitesimal Stream b, Eq a, Universe a) => VectorDerivative (a -> b) Dual LinearMap Source #
Instance details Methods divergence :: LinearMap (a -> b) (a -> b) -> Dual (a -> b) Source # grad :: Dual (a -> b) -> LinearMap (a -> b) (a -> b) Source # directional_derivative :: (a -> b) -> Dual (a -> b) -> Dual (a -> b) Source # laplace :: Dual (a -> b) -> Dual (a -> b) Source #
(Integral col, Floating elem, Scalar (col -> elem) ~ elem, VectorSpace elem, Fractional (Scalar elem), Integral row, Universe row, Universe col, ConjugateSymmetric elem) => Num ((row :&: col) elem) Source #
Instance details Methods (+) :: (row :&: col) elem -> (row :&: col) elem -> (row :&: col) elem # (-) :: (row :&: col) elem -> (row :&: col) elem -> (row :&: col) elem # (*) :: (row :&: col) elem -> (row :&: col) elem -> (row :&: col) elem # negate :: (row :&: col) elem -> (row :&: col) elem # abs :: (row :&: col) elem -> (row :&: col) elem # signum :: (row :&: col) elem -> (row :&: col) elem # fromInteger :: Integer -> (row :&: col) elem #
(Integral row, Universe row, SupportsMatrixMultiplication ((->) row) ((->) row) ((->) row) a) => LieAlgebra ((row :&: row) a) Source #
Instance details Methods (%<>%) :: (row :&: row) a -> (row :&: row) a -> (row :&: row) a Source #
(VectorSpace a, Integral col, Integral row, Universe row, ConjugateSymmetric a, Universe col, Floating a, Integral row, Scalar (col -> a) ~ a) => NormedSpace ((row :&: col) a) Source #
Instance details Methods norm :: (row :&: col) a -> Scalar ((row :&: col) a) Source # norm_squared :: (row :&: col) a -> Scalar ((row :&: col) a) Source #
Eq ind => UpdateableMatrixDimension ((->) ind) Source #
Instance details Methods write_row :: Applicative h => h a -> ind -> (((->) ind :: h) a -> ((->) ind :: h) a) Source # write_column :: Applicative h => h a -> ind -> ((h :: (->) ind) a -> (h :: (->) ind) a) Source #
Comonad ((->) FiveD) Source #
Instance details Methods extract :: (FiveD -> a) -> a Source # duplicate :: (FiveD -> a) -> FiveD -> (FiveD -> a) Source # extend :: ((FiveD -> a) -> b) -> (FiveD -> a) -> FiveD -> b Source # (=>>) :: (FiveD -> a) -> ((FiveD -> a) -> b) -> FiveD -> b Source # (.>>) :: (FiveD -> a) -> b -> FiveD -> b Source #
Comonad ((->) FourD) Source #
Instance details Methods extract :: (FourD -> a) -> a Source # duplicate :: (FourD -> a) -> FourD -> (FourD -> a) Source # extend :: ((FourD -> a) -> b) -> (FourD -> a) -> FourD -> b Source # (=>>) :: (FourD -> a) -> ((FourD -> a) -> b) -> FourD -> b Source # (.>>) :: (FourD -> a) -> b -> FourD -> b Source #
Comonad ((->) SixD) Source #
Instance details Methods extract :: (SixD -> a) -> a Source # duplicate :: (SixD -> a) -> SixD -> (SixD -> a) Source # extend :: ((SixD -> a) -> b) -> (SixD -> a) -> SixD -> b Source # (=>>) :: (SixD -> a) -> ((SixD -> a) -> b) -> SixD -> b Source # (.>>) :: (SixD -> a) -> b -> SixD -> b Source #
Comonad ((->) ThreeD) Source #
Instance details Methods extract :: (ThreeD -> a) -> a Source # duplicate :: (ThreeD -> a) -> ThreeD -> (ThreeD -> a) Source # extend :: ((ThreeD -> a) -> b) -> (ThreeD -> a) -> ThreeD -> b Source # (=>>) :: (ThreeD -> a) -> ((ThreeD -> a) -> b) -> ThreeD -> b Source # (.>>) :: (ThreeD -> a) -> b -> ThreeD -> b Source #
Comonad ((->) TwoD) Source #
Instance details Methods extract :: (TwoD -> a) -> a Source # duplicate :: (TwoD -> a) -> TwoD -> (TwoD -> a) Source # extend :: ((TwoD -> a) -> b) -> (TwoD -> a) -> TwoD -> b Source # (=>>) :: (TwoD -> a) -> ((TwoD -> a) -> b) -> TwoD -> b Source # (.>>) :: (TwoD -> a) -> b -> TwoD -> b Source #
Universe a => PpShowF ((->) a) Source #
Instance details Methods ppf :: PpShow a0 => (a -> a0) -> Doc Source #
Universe a => PpShowVerticalF ((->) a) Source #
Instance details Methods ppf_vertical :: PpShow a0 => (a -> a0) -> Doc Source #
(Num a, Eq dim, Integral dim) => Diagonalizable ((->) dim) a Source #
Instance details Methods identity_impl :: (dim -> Integer) -> ((->) dim :: (->) dim) a Source # identity :: ((->) dim :: (->) dim) a Source # diagonal_impl :: ((->) dim :: (->) dim) a -> dim -> a Source # diagonal_matrix_impl :: (dim -> a) -> ((->) dim :: (->) dim) a Source #
(Num a, Integral row) => Indexable ((->) row) a Source #
Instance details Methods diagonal_projections :: row -> Index ((->) row) a Source # basis_vector :: Index ((->) row) a -> row -> a Source # index_project :: Index ((->) row) a -> (row -> a) -> a Source # indexable_indices :: row -> a Source #
(Eq dim, Num dim, Integral dim, Universe dim, CoFactorDimension dim, ConjugateSymmetric a, Num a) => Traceable ((->) dim) a Source #
Instance details Methods trace_impl :: ((->) dim :: (->) dim) a -> a Source # determinant_impl :: ((->) dim :: (->) dim) a -> a Source # vector_dimension :: (dim -> a) -> a Source #
(Universe row, Num a) => LinearTransform ((->) row) Vector1 a Source #
Instance details Methods (<>>) :: Vector1 a -> ((->) row :: Vector1) a -> row -> a Source # (<<>) :: ((->) row :: Vector1) a -> (row -> a) -> Vector1 a Source #
(Universe row, Num a) => LinearTransform ((->) row) Vector2 a Source #
Instance details Methods (<>>) :: Vector2 a -> ((->) row :: Vector2) a -> row -> a Source # (<<>) :: ((->) row :: Vector2) a -> (row -> a) -> Vector2 a Source #
(Universe row, Num a) => LinearTransform ((->) row) Vector3 a Source #
Instance details Methods (<>>) :: Vector3 a -> ((->) row :: Vector3) a -> row -> a Source # (<<>) :: ((->) row :: Vector3) a -> (row -> a) -> Vector3 a Source #
(Universe row, Num a) => LinearTransform ((->) row) Vector4 a Source #
Instance details Methods (<>>) :: Vector4 a -> ((->) row :: Vector4) a -> row -> a Source # (<<>) :: ((->) row :: Vector4) a -> (row -> a) -> Vector4 a Source #
(Num a, Universe row, Universe col) => LinearTransform ((->) row) ((->) col) a Source #
Instance details Methods (<>>) :: (col -> a) -> ((->) row :: (->) col) a -> row -> a Source # (<<>) :: ((->) row :: (->) col) a -> (row -> a) -> col -> a Source #
(Integral col, Integral row, Universe row, Universe col, Num a, ConjugateSymmetric a) => InnerProductSpace (((->) row :*: (->) col) a) Source #	https://en.wikipedia.org/wiki/Frobenius_inner_product
Instance details Methods (%.) :: ((->) row :: (->) col) a -> ((->) row :: (->) col) a -> Scalar (((->) row :*: (->) col) a) Source #
(Integral row, Integral col, Floating a, Universe row, Universe col, ConjugateSymmetric a) => NormedSpace (((->) row :*: (->) col) a) Source #
Instance details Methods norm :: ((->) row :: (->) col) a -> Scalar (((->) row :: (->) col) a) Source # norm_squared :: ((->) row :: (->) col) a -> Scalar (((->) row :: (->) col) a) Source #
Num a => VectorSpace ((Vector1 :*: (->) col) a) Source #
Instance details Associated Types type Scalar ((Vector1 :: (->) col) a) Source # Methods vzero :: (Vector1 :: (->) col) a Source # vnegate :: (Vector1 :: (->) col) a -> (Vector1 :: (->) col) a Source # (%+) :: (Vector1 :: (->) col) a -> (Vector1 :: (->) col) a -> (Vector1 :: (->) col) a Source # (%) :: Scalar ((Vector1 :: (->) col) a) -> (Vector1 :: (->) col) a -> (Vector1 :*: (->) col) a Source #
Num a => VectorSpace ((Vector2 :*: (->) col) a) Source #
Instance details Associated Types type Scalar ((Vector2 :: (->) col) a) Source # Methods vzero :: (Vector2 :: (->) col) a Source # vnegate :: (Vector2 :: (->) col) a -> (Vector2 :: (->) col) a Source # (%+) :: (Vector2 :: (->) col) a -> (Vector2 :: (->) col) a -> (Vector2 :: (->) col) a Source # (%) :: Scalar ((Vector2 :: (->) col) a) -> (Vector2 :: (->) col) a -> (Vector2 :*: (->) col) a Source #
Num a => VectorSpace ((Vector3 :*: (->) col) a) Source #
Instance details Associated Types type Scalar ((Vector3 :: (->) col) a) Source # Methods vzero :: (Vector3 :: (->) col) a Source # vnegate :: (Vector3 :: (->) col) a -> (Vector3 :: (->) col) a Source # (%+) :: (Vector3 :: (->) col) a -> (Vector3 :: (->) col) a -> (Vector3 :: (->) col) a Source # (%) :: Scalar ((Vector3 :: (->) col) a) -> (Vector3 :: (->) col) a -> (Vector3 :*: (->) col) a Source #
Num a => VectorSpace ((Vector4 :*: (->) col) a) Source #
Instance details Associated Types type Scalar ((Vector4 :: (->) col) a) Source # Methods vzero :: (Vector4 :: (->) col) a Source # vnegate :: (Vector4 :: (->) col) a -> (Vector4 :: (->) col) a Source # (%+) :: (Vector4 :: (->) col) a -> (Vector4 :: (->) col) a -> (Vector4 :: (->) col) a Source # (%) :: Scalar ((Vector4 :: (->) col) a) -> (Vector4 :: (->) col) a -> (Vector4 :*: (->) col) a Source #
Num a => VectorSpace (((->) row :*: Vector1) a) Source #
Instance details Associated Types type Scalar (((->) row :: Vector1) a) Source # Methods vzero :: ((->) row :: Vector1) a Source # vnegate :: ((->) row :: Vector1) a -> ((->) row :: Vector1) a Source # (%+) :: ((->) row :: Vector1) a -> ((->) row :: Vector1) a -> ((->) row :: Vector1) a Source # (%) :: Scalar (((->) row :: Vector1) a) -> ((->) row :: Vector1) a -> ((->) row :*: Vector1) a Source #
Num a => VectorSpace (((->) row :*: Vector2) a) Source #
Instance details Associated Types type Scalar (((->) row :: Vector2) a) Source # Methods vzero :: ((->) row :: Vector2) a Source # vnegate :: ((->) row :: Vector2) a -> ((->) row :: Vector2) a Source # (%+) :: ((->) row :: Vector2) a -> ((->) row :: Vector2) a -> ((->) row :: Vector2) a Source # (%) :: Scalar (((->) row :: Vector2) a) -> ((->) row :: Vector2) a -> ((->) row :*: Vector2) a Source #
Num a => VectorSpace (((->) row :*: Vector3) a) Source #
Instance details Associated Types type Scalar (((->) row :: Vector3) a) Source # Methods vzero :: ((->) row :: Vector3) a Source # vnegate :: ((->) row :: Vector3) a -> ((->) row :: Vector3) a Source # (%+) :: ((->) row :: Vector3) a -> ((->) row :: Vector3) a -> ((->) row :: Vector3) a Source # (%) :: Scalar (((->) row :: Vector3) a) -> ((->) row :: Vector3) a -> ((->) row :*: Vector3) a Source #
Num a => VectorSpace (((->) row :*: Vector4) a) Source #
Instance details Associated Types type Scalar (((->) row :: Vector4) a) Source # Methods vzero :: ((->) row :: Vector4) a Source # vnegate :: ((->) row :: Vector4) a -> ((->) row :: Vector4) a Source # (%+) :: ((->) row :: Vector4) a -> ((->) row :: Vector4) a -> ((->) row :: Vector4) a Source # (%) :: Scalar (((->) row :: Vector4) a) -> ((->) row :: Vector4) a -> ((->) row :*: Vector4) a Source #
Num a => VectorSpace (((->) row :*: (->) col) a) Source #
Instance details Associated Types type Scalar (((->) row :: (->) col) a) Source # Methods vzero :: ((->) row :: (->) col) a Source # vnegate :: ((->) row :: (->) col) a -> ((->) row :: (->) col) a Source # (%+) :: ((->) row :: (->) col) a -> ((->) row :: (->) col) a -> ((->) row :: (->) col) a Source # (%) :: Scalar (((->) row :: (->) col) a) -> ((->) row :: (->) col) a -> ((->) row :*: (->) col) a Source #