Copyright	(c) Esa Pulkkinen 2018
License	LGPL
Maintainer	esa.pulkkinen@iki.fi
Safe Haskell	Safe
Language	Haskell2010

Math.Matrix.Covector

Contents

Orphan instances

Description

For background material, see K. Chandrasekhara Rao: Functional Analysis.

Synopsis

contract :: (Scalar (v a) ~ a, LinearTransform v Vector1 a) => (Dual :*: v) a -> (Vector1 :*: v) a -> Scalar (v a)
dual_apply :: LinearTransform v Vector1 a => Dual (v a) -> v a -> a
(*><) :: a ~ Scalar (v a) => Dual (v a) -> v a :-> Vector1 a
x2_op :: (Num s, ConjugateSymmetric s) => Dual (Vector2 s)
y2_op :: (Num s, ConjugateSymmetric s) => Dual (Vector2 s)
(∇) :: VectorDerivative v d arr => d v -> arr v v
(∇·) :: VectorDerivative v d arr => arr v v -> d v
(·∇) :: (VectorDerivative v d arr, InnerProductSpace v) => v -> d v -> d v
(∇×) :: VectorCrossProduct v arr => arr v v -> arr v v
(∇·∇) :: VectorDerivative v d arr => d v -> d v
(∇×∇×) :: VectorCrossProduct v arr => arr v v -> arr v v
directional_derivative_endo :: (VectorDerivative v Dual LinearMap, InnerProductSpace v) => v -> Endo (Dual v)
dconst :: (Scalar (f v) ~ v, Dualizable (f v) d, Linearizable LinearMap (:*:) f Vector1 v, Diagonalizable f v) => v -> d (f v)
outer_vector :: (Linearizable LinearMap (:*:) f g w, DualNum f w, DualNum g w) => Dual (f w) -> g w -> f w :-> g w
conjugate_transpose :: (Diagonalizable m a, Linearizable LinearMap (:*:) m m a, Linearizable LinearMap (:*:) m n a, Linearizable LinearMap (:*:) n m a, Diagonalizable n a, LinearIso n m a, Transposable m n a, ConjugateSymmetric (m a)) => (m a :-> n a) -> n a :-> m a
(⊹) :: (Linearizable LinearMap (:*:) m m a, Linearizable LinearMap (:*:) m n a, Linearizable LinearMap (:*:) n m a, Diagonalizable m a, Diagonalizable n a, LinearIso n m a, Transposable m n a, ConjugateSymmetric (m a)) => (m a :-> n a) -> n a :-> m a
is_hermitian :: (Eq (m (m a)), ConjugateSymmetric (m a), Linearizable LinearMap (:*:) m m a, Transposable m m a, Diagonalizable m a, LinearTransform m m a, Applicative m, Foldable m) => (m a :-> m a) -> Bool
is_skew_hermitian :: (Num (Scalar (m a)), Eq (m (m a)), ConjugateSymmetric (m a), Applicative m, Linearizable LinearMap (:*:) m m a, VectorSpace (m a), Diagonalizable m a, LinearTransform m m a, Transposable m m a, Foldable m) => (m a :-> m a) -> Bool
is_normal :: (Eq (m (m a)), Diagonalizable m a, LinearTransform m m a, Linearizable LinearMap (:*:) m m a, ConjugateSymmetric (m a)) => (m a :-> m a) -> Bool
bravector :: (DualNum f v, InnerProductSpace (f v)) => f v -> Dual (f v)
ketvector :: (DualNum f v, InnerProductSpace (f v)) => f v -> Dual (f v)
kronecker :: (Eq i, Functor n, Functor m, Num c) => m i -> n i -> (m :*: n) c
dual_bilinear :: DualNum f v => (f v -> f v -> v) -> f v -> Dual (f v)
kronecker_dual :: (DualNum f a, Eq (f a), Num a) => f a -> Dual (f a)
covector_kernel :: (DualNum f v, Num v, Eq v) => Dual (f v) -> f v -> Bool
adjoint :: (f a :-> f a) -> Dual (f a) -> Dual (f a)
scalar_map :: a ~ Scalar (f a) => (Vector1 a :-> Vector1 a) -> Dual (f a) -> Dual (f a)
operator_map :: (Scalar (w a) ~ a, Dualizable (w a) Dual, Dualizable (v a) Dual, Diagonalizable w a, LinearTransform v Vector1 a, Linearizable LinearMap (:*:) w Vector1 a) => ((v a -> a) -> w a -> a) -> Dual (v a) -> Dual (w a)
type DualNum f v = (LinearTransform f Vector1 v, Dualizable (f v) Dual, ConjugateSymmetric v, Diagonalizable f v, Linearizable LinearMap (:*:) f Vector1 v, VectorSpace (f v))
dual_derivative :: (VectorDerivative v Dual LinearMap, VectorDerivative w Dual LinearMap, InnerProductSpace v, InnerProductSpace w) => (Dual v -> Dual w -> c) -> v -> w -> Dual v -> Dual w -> c
dual_differential_outer_product :: (Dualizable (f v) Dual, LinearTransform f Vector1 v, ConjugateSymmetric v, Diagonalizable f v, Linearizable LinearMap (:*:) f Vector1 v, VectorDerivative (f v) Dual LinearMap, InnerProductSpace (f v)) => f v -> f v -> Dual (f v) -> Dual (f v) -> Dual (f v)
dual_differential_dot_product :: (DualNum f v, VectorDerivative (f v) Dual LinearMap, InnerProductSpace (f v), StandardBasis (f v)) => f v -> f v -> Dual (f v) -> Dual (f v) -> Scalar (Dual (f v))
norm_covector :: (Diagonalizable f v, Linearizable LinearMap (:*:) f Vector1 v, Dualizable (f v) d, NormedSpace (f v)) => d (f v)
del2 :: (ConjugateSymmetric v, Closed v) => Vector2 (Dual (Vector2 v) -> Dual (Vector2 v))
hessian2 :: (ConjugateSymmetric a, Closed a) => Dual (Vector2 a) -> (Vector2 :*: Vector2) (Dual (Vector2 a))

Documentation

contract :: (Scalar (v a) ~ a, LinearTransform v Vector1 a) => (Dual :*: v) a -> (Vector1 :*: v) a -> Scalar (v a) Source #

dual_apply :: LinearTransform v Vector1 a => Dual (v a) -> v a -> a Source #

(*><) :: a ~ Scalar (v a) => Dual (v a) -> v a :-> Vector1 a Source #

x2_op :: (Num s, ConjugateSymmetric s) => Dual (Vector2 s) Source #

y2_op :: (Num s, ConjugateSymmetric s) => Dual (Vector2 s) Source #

(∇) :: VectorDerivative v d arr => d v -> arr v v Source #

unicode support for gradient. Use with parenthesis.

(∇·) :: VectorDerivative v d arr => arr v v -> d v Source #

unicode support for divergence. Use with parenthesis.

(·∇) :: (VectorDerivative v d arr, InnerProductSpace v) => v -> d v -> d v Source #

unicode support for directional derivative. Binary operator.

(∇×) :: VectorCrossProduct v arr => arr v v -> arr v v Source #

unicode support for curl. Use with parenthesis.

(∇·∇) :: VectorDerivative v d arr => d v -> d v Source #

unicode support for laplace. Use with parenthesis.

(∇×∇×) :: VectorCrossProduct v arr => arr v v -> arr v v Source #

directional_derivative_endo :: (VectorDerivative v Dual LinearMap, InnerProductSpace v) => v -> Endo (Dual v) Source #

dconst :: (Scalar (f v) ~ v, Dualizable (f v) d, Linearizable LinearMap (:*:) f Vector1 v, Diagonalizable f v) => v -> d (f v) Source #

outer_vector :: (Linearizable LinearMap (:*:) f g w, DualNum f w, DualNum g w) => Dual (f w) -> g w -> f w :-> g w Source #

conjugate_transpose :: (Diagonalizable m a, Linearizable LinearMap (:*:) m m a, Linearizable LinearMap (:*:) m n a, Linearizable LinearMap (:*:) n m a, Diagonalizable n a, LinearIso n m a, Transposable m n a, ConjugateSymmetric (m a)) => (m a :-> n a) -> n a :-> m a Source #

http://en.wikipedia.org/wiki/Conjugate_transpose

(⊹) :: (Linearizable LinearMap (:*:) m m a, Linearizable LinearMap (:*:) m n a, Linearizable LinearMap (:*:) n m a, Diagonalizable m a, Diagonalizable n a, LinearIso n m a, Transposable m n a, ConjugateSymmetric (m a)) => (m a :-> n a) -> n a :-> m a Source #

unicode alias (unicode HERMITIAN CONJUGATE MATRIX character)

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

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

is_skew_hermitian :: (Num (Scalar (m a)), Eq (m (m a)), ConjugateSymmetric (m a), Applicative m, Linearizable LinearMap (:*:) m m a, VectorSpace (m a), Diagonalizable m a, LinearTransform m m a, Transposable m m a, Foldable m) => (m a :-> m a) -> Bool Source #

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

is_normal :: (Eq (m (m a)), Diagonalizable m a, LinearTransform m m a, Linearizable LinearMap (:*:) m m a, ConjugateSymmetric (m a)) => (m a :-> m a) -> Bool Source #

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

bravector :: (DualNum f v, InnerProductSpace (f v)) => f v -> Dual (f v) Source #

partial application of dot product on second argument (- %. v).

ketvector :: (DualNum f v, InnerProductSpace (f v)) => f v -> Dual (f v) Source #

partial application of dot product on first argument: (v %. -).

kronecker :: (Eq i, Functor n, Functor m, Num c) => m i -> n i -> (m :*: n) c Source #

dual_bilinear :: DualNum f v => (f v -> f v -> v) -> f v -> Dual (f v) Source #

kronecker_dual :: (DualNum f a, Eq (f a), Num a) => f a -> Dual (f a) Source #

covector_kernel :: (DualNum f v, Num v, Eq v) => Dual (f v) -> f v -> Bool Source #

https://en.wikipedia.org/wiki/Dual_space#Injection_into_the_double-dual

adjoint :: (f a :-> f a) -> Dual (f a) -> Dual (f a) Source #

would like to make adjoint as instance of CoFunctor, but the constraint prevents. Instead we declare LinearMap.

scalar_map :: a ~ Scalar (f a) => (Vector1 a :-> Vector1 a) -> Dual (f a) -> Dual (f a) Source #

operator_map :: (Scalar (w a) ~ a, Dualizable (w a) Dual, Dualizable (v a) Dual, Diagonalizable w a, LinearTransform v Vector1 a, Linearizable LinearMap (:*:) w Vector1 a) => ((v a -> a) -> w a -> a) -> Dual (v a) -> Dual (w a) Source #

type DualNum f v = (LinearTransform f Vector1 v, Dualizable (f v) Dual, ConjugateSymmetric v, Diagonalizable f v, Linearizable LinearMap (:*:) f Vector1 v, VectorSpace (f v)) Source #

dual_derivative :: (VectorDerivative v Dual LinearMap, VectorDerivative w Dual LinearMap, InnerProductSpace v, InnerProductSpace w) => (Dual v -> Dual w -> c) -> v -> w -> Dual v -> Dual w -> c Source #

This computes \(DDM(\otimes,{\bar{x}},{\bar{y}}) = [({\bar{x}} \cdot \nabla) \times ({\bar{y}} \cdot \nabla)]\) where \(\times\) is the outer product for linear operators derived from product of dual spaces \(\otimes\). Notice \({\bar{x}} \cdot \nabla\) is normally called the directional derivative, so this is really a two dimensional version of directional derivative with some possibility to choose how two dual vectors are combined.

dual_differential_outer_product :: (Dualizable (f v) Dual, LinearTransform f Vector1 v, ConjugateSymmetric v, Diagonalizable f v, Linearizable LinearMap (:*:) f Vector1 v, VectorDerivative (f v) Dual LinearMap, InnerProductSpace (f v)) => f v -> f v -> Dual (f v) -> Dual (f v) -> Dual (f v) Source #

dual_differential_dot_product :: (DualNum f v, VectorDerivative (f v) Dual LinearMap, InnerProductSpace (f v), StandardBasis (f v)) => f v -> f v -> Dual (f v) -> Dual (f v) -> Scalar (Dual (f v)) Source #

norm_covector :: (Diagonalizable f v, Linearizable LinearMap (:*:) f Vector1 v, Dualizable (f v) d, NormedSpace (f v)) => d (f v) Source #

del2 :: (ConjugateSymmetric v, Closed v) => Vector2 (Dual (Vector2 v) -> Dual (Vector2 v)) Source #

hessian2 :: (ConjugateSymmetric a, Closed a) => Dual (Vector2 a) -> (Vector2 :*: Vector2) (Dual (Vector2 a)) Source #

Orphan instances

(Dualizable (f v) Dual, LinearTransform f Vector1 v, ConjugateSymmetric v, Linearizable LinearMap ((:*:) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) f Vector1 v, FractionalSpace (f v), v ~ Scalar (f v), Floating v, Diagonalizable f v) => Floating (Dual (f v)) Source #
Instance details Methods pi :: Dual (f v) # exp :: Dual (f v) -> Dual (f v) # log :: Dual (f v) -> Dual (f v) # sqrt :: Dual (f v) -> Dual (f v) # (**) :: Dual (f v) -> Dual (f v) -> Dual (f v) # logBase :: Dual (f v) -> Dual (f v) -> Dual (f v) # sin :: Dual (f v) -> Dual (f v) # cos :: Dual (f v) -> Dual (f v) # tan :: Dual (f v) -> Dual (f v) # asin :: Dual (f v) -> Dual (f v) # acos :: Dual (f v) -> Dual (f v) # atan :: Dual (f v) -> Dual (f v) # sinh :: Dual (f v) -> Dual (f v) # cosh :: Dual (f v) -> Dual (f v) # tanh :: Dual (f v) -> Dual (f v) # asinh :: Dual (f v) -> Dual (f v) # acosh :: Dual (f v) -> Dual (f v) # atanh :: Dual (f v) -> Dual (f v) # log1p :: Dual (f v) -> Dual (f v) # expm1 :: Dual (f v) -> Dual (f v) # log1pexp :: Dual (f v) -> Dual (f v) # log1mexp :: Dual (f v) -> Dual (f v) #
DualNum f v => Num (Dual (f v)) Source #
Instance details Methods (+) :: Dual (f v) -> Dual (f v) -> Dual (f v) # (-) :: Dual (f v) -> Dual (f v) -> Dual (f v) # (*) :: Dual (f v) -> Dual (f v) -> Dual (f v) # negate :: Dual (f v) -> Dual (f v) # abs :: Dual (f v) -> Dual (f v) # signum :: Dual (f v) -> Dual (f v) # fromInteger :: Integer -> Dual (f v) #
(DualNum f v, FractionalSpace (f v)) => Fractional (Dual (f v)) Source #
Instance details Methods (/) :: Dual (f v) -> Dual (f v) -> Dual (f v) # recip :: Dual (f v) -> Dual (f v) # fromRational :: Rational -> Dual (f v) #
(DualNum f v, StandardBasis (f v), Show v) => Show (Dual (f v)) Source #
Instance details Methods showsPrec :: Int -> Dual (f v) -> ShowS # show :: Dual (f v) -> String # showList :: [Dual (f v)] -> ShowS #
(StandardBasis (f v), Num v, VectorSpace (f v), DualNum f v) => InnerProductSpace (Dual (f v)) Source #
Instance details Methods (%.) :: Dual (f v) -> Dual (f v) -> Scalar (Dual (f v)) Source #
(v ~ Scalar (f v), VectorSpace (f v), DualNum f v) => LieAlgebra (Dual (f v)) Source #
Instance details Methods (%<>%) :: Dual (f v) -> Dual (f v) -> Dual (f v) Source #
(Semigroup v, VectorSpace v) => Monoid (LinearMap v v) Source #
Instance details Methods mempty :: LinearMap v v # mappend :: LinearMap v v -> LinearMap v v -> LinearMap v v # mconcat :: [LinearMap v v] -> LinearMap v v #
Semigroup v => Semigroup (LinearMap v v) Source #
Instance details Methods (<>) :: LinearMap v v -> LinearMap v v -> LinearMap v v # sconcat :: NonEmpty (LinearMap v v) -> LinearMap v v # stimes :: Integral b => b -> LinearMap v v -> LinearMap v v #
(ConjugateSymmetric a, Num a) => StandardBasis ((Dual :*: Vector2) a) Source #	https://en.wikipedia.org/wiki/Dual_space
Instance details Methods unit_vectors :: [(Dual :*: Vector2) a] Source #