Copyright	(c) Ross Paterson 2011
License	BSD-style (see the file LICENSE)
Maintainer	[email protected]
Stability	provisional
Portability	portable
Safe Haskell	None
Language	Haskell2010

Data.YAP.WeylAlgebra

Contents

Weyl algebras
Construction
Queries

Description

An example instance of the algebraic classes: Weyl algebras.

Synopsis

data Weyl a
fromPolynomial :: Polynomial a -> Weyl a
diff :: Semiring a => Weyl a
fromCoefficients :: [Polynomial a] -> Weyl a
order :: (Eq a, AdditiveMonoid a) => Weyl a -> Int
coefficients :: (Eq a, AdditiveMonoid a) => Weyl a -> [Polynomial a]
evaluate :: Semiring a => Weyl a -> Polynomial a -> Polynomial a
evaluateWith :: (Eq a, AdditiveMonoid a, Differentiable b) => Weyl a -> (a -> b) -> b -> b -> b

Weyl algebras

data Weyl a Source #

The (first) Weyl algebra over a, consisting of differential operators of the form \[ p_0(x) + p_1(x) \partial_x + \cdots + p_n(x) \partial_x^n \] where each \(p_i(x)\) is a polynomial in \(x\) and each \(\partial_x^i\) is the differentiation operator with respect to \(x\) repeated \(i\) times.

Instances

Instances details

AdditiveFunctor Weyl Source #
Instance details Defined in Data.YAP.WeylAlgebra Methods mapAdditive :: (AdditiveMonoid a, AdditiveMonoid b) => (a -> b) -> Weyl a -> Weyl b #
(Eq a, Show a, AdditiveMonoid a) => Show (Weyl a) Source #
Instance details Defined in Data.YAP.WeylAlgebra Methods showsPrec :: Int -> Weyl a -> ShowS # show :: Weyl a -> String # showList :: [Weyl a] -> ShowS #
(Eq a, AdditiveMonoid a) => Eq (Weyl a) Source #
Instance details Defined in Data.YAP.WeylAlgebra Methods (==) :: Weyl a -> Weyl a -> Bool # (/=) :: Weyl a -> Weyl a -> Bool #
(Ord a, AdditiveMonoid a) => Ord (Weyl a) Source #
Instance details Defined in Data.YAP.WeylAlgebra Methods compare :: Weyl a -> Weyl a -> Ordering # (<) :: Weyl a -> Weyl a -> Bool # (<=) :: Weyl a -> Weyl a -> Bool # (>) :: Weyl a -> Weyl a -> Bool # (>=) :: Weyl a -> Weyl a -> Bool # max :: Weyl a -> Weyl a -> Weyl a # min :: Weyl a -> Weyl a -> Weyl a #
AbelianGroup a => AbelianGroup (Weyl a) Source #	Pointwise negation
Instance details Defined in Data.YAP.WeylAlgebra Methods (-) :: Weyl a -> Weyl a -> Weyl a # negate :: Weyl a -> Weyl a # gtimes :: (AbelianGroup b, ToInteger b) => b -> Weyl a -> Weyl a #
AdditiveMonoid a => AdditiveMonoid (Weyl a) Source #	Pointwise addition
Instance details Defined in Data.YAP.WeylAlgebra Methods (+) :: Weyl a -> Weyl a -> Weyl a # zero :: Weyl a # atimes :: ToInteger b => b -> Weyl a -> Weyl a #
FromRational a => FromRational (Weyl a) Source #
Instance details Defined in Data.YAP.WeylAlgebra Methods fromRational :: Rational -> Weyl a #
Ring a => Ring (Weyl a) Source #
Instance details Defined in Data.YAP.WeylAlgebra Methods fromInteger :: Integer -> Weyl a #
Semiring a => Semiring (Weyl a) Source #	Composition of operators
Instance details Defined in Data.YAP.WeylAlgebra Methods (*) :: Weyl a -> Weyl a -> Weyl a # one :: Weyl a # fromNatural :: Natural -> Weyl a # rescale :: Weyl a -> Weyl a -> (Weyl a, Weyl a, Weyl a -> Weyl a) #

Construction

fromPolynomial :: Polynomial a -> Weyl a Source #

Operator representing multiplication by a polynomial

diff :: Semiring a => Weyl a Source #

Derivative operator, i.e. \(\partial_x\), satisfying

diff*fromPolynomial p = fromPolynomial p*diff + fromPolynomial (derivative p)

fromCoefficients :: [Polynomial a] -> Weyl a Source #

Construct an operator from a list of coefficients of the iterated differential operators in order of increasing number of iterated derivatives.

Queries

order :: (Eq a, AdditiveMonoid a) => Weyl a -> Int Source #

The order of a differential operator.

coefficients :: (Eq a, AdditiveMonoid a) => Weyl a -> [Polynomial a] Source #

Coefficients of the iterated differential operators in order of increasing number of iterations, and with no trailing zeros.

evaluate :: Semiring a => Weyl a -> Polynomial a -> Polynomial a Source #

Evaluate an operator on a given polynomial.

evaluateWith :: (Eq a, AdditiveMonoid a, Differentiable b) => Weyl a -> (a -> b) -> b -> b -> b Source #

Evaluate an operator, given embeddings of values and the identity polynomial.