Copyright	(c) Ross Paterson 2021
License	BSD-style (see the file LICENSE)
Maintainer	[email protected]
Stability	provisional
Portability	portable
Safe Haskell	None
Language	Haskell2010

Data.YAP.Polynomial.Z2

Contents

Description

An example instance of the algebraic classes: formal polynomials specialized to \({\Bbb Z}_2\), which can be represented as bit sequences.

Synopsis

data PolynomialZ2 a
fromBits :: a -> PolynomialZ2 a
fromIndices :: Bits a => [Int] -> PolynomialZ2 a
degree :: Bits a => PolynomialZ2 a -> Int
bits :: PolynomialZ2 a -> a
indices :: Bits a => PolynomialZ2 a -> [Int]
evaluate :: (Bits a, Semiring b) => PolynomialZ2 a -> b -> b
identity :: Bits a => PolynomialZ2 a
compose :: Bits a => PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a

Polynomials

data PolynomialZ2 a Source #

Polynomial over \({\Bbb Z}_2\), with coefficients represented by the bits of a value of type a.

Instances

Instances details

Show a => Show (PolynomialZ2 a) Source #
Instance details Defined in Data.YAP.Polynomial.Z2 Methods showsPrec :: Int -> PolynomialZ2 a -> ShowS # show :: PolynomialZ2 a -> String # showList :: [PolynomialZ2 a] -> ShowS #
Eq a => Eq (PolynomialZ2 a) Source #
Instance details Defined in Data.YAP.Polynomial.Z2 Methods (==) :: PolynomialZ2 a -> PolynomialZ2 a -> Bool # (/=) :: PolynomialZ2 a -> PolynomialZ2 a -> Bool #
Ord a => Ord (PolynomialZ2 a) Source #
Instance details Defined in Data.YAP.Polynomial.Z2 Methods compare :: PolynomialZ2 a -> PolynomialZ2 a -> Ordering # (<) :: PolynomialZ2 a -> PolynomialZ2 a -> Bool # (<=) :: PolynomialZ2 a -> PolynomialZ2 a -> Bool # (>) :: PolynomialZ2 a -> PolynomialZ2 a -> Bool # (>=) :: PolynomialZ2 a -> PolynomialZ2 a -> Bool # max :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a # min :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a #
Bits a => AbelianGroup (PolynomialZ2 a) Source #	Negation is the identity.
Instance details Defined in Data.YAP.Polynomial.Z2 Methods (-) :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a # negate :: PolynomialZ2 a -> PolynomialZ2 a # gtimes :: (AbelianGroup b, ToInteger b) => b -> PolynomialZ2 a -> PolynomialZ2 a #
Bits a => AdditiveMonoid (PolynomialZ2 a) Source #	Addition without carrying, i.e. exclusive or.
Instance details Defined in Data.YAP.Polynomial.Z2 Methods (+) :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a # zero :: PolynomialZ2 a # atimes :: ToInteger b => b -> PolynomialZ2 a -> PolynomialZ2 a #
Bits a => Euclidean (PolynomialZ2 a) Source #
Instance details Defined in Data.YAP.Polynomial.Z2 Methods div :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a # mod :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a # divMod :: PolynomialZ2 a -> PolynomialZ2 a -> (PolynomialZ2 a, PolynomialZ2 a) # euclideanNorm :: PolynomialZ2 a -> Natural #
Bits a => Ring (PolynomialZ2 a) Source #	`fromInteger n` is `0` if `n` is even and `1` if it is odd.
Instance details Defined in Data.YAP.Polynomial.Z2 Methods fromInteger :: Integer -> PolynomialZ2 a #
Bits a => Semiring (PolynomialZ2 a) Source #	Multiplication without carrying. `fromNatural n` is `0` if `n` is even and `1` if it is odd.
Instance details Defined in Data.YAP.Polynomial.Z2 Methods (*) :: PolynomialZ2 a -> PolynomialZ2 a -> PolynomialZ2 a # one :: PolynomialZ2 a # fromNatural :: Natural -> PolynomialZ2 a # rescale :: PolynomialZ2 a -> PolynomialZ2 a -> (PolynomialZ2 a, PolynomialZ2 a, PolynomialZ2 a -> PolynomialZ2 a) #
Bits a => StandardAssociate (PolynomialZ2 a) Source #	The only unit is `1`. `stdAssociate` is the identity.
Instance details Defined in Data.YAP.Polynomial.Z2 Methods stdAssociate :: PolynomialZ2 a -> PolynomialZ2 a # stdUnit :: PolynomialZ2 a -> PolynomialZ2 a # stdRecip :: PolynomialZ2 a -> PolynomialZ2 a #