HaskellForMaths-0.4.9: Combinatorics, group theory, commutative algebra, non-commutative algebra
Safe HaskellNone
LanguageHaskell98

Math.Algebra.NonCommutative.NCPoly

Description

A module providing a type for non-commutative polynomials.

Synopsis

Documentation

newtype Monomial v Source #

Constructors

M [v] 

Instances

Instances details
Eq v => Eq (Monomial v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

(==) :: Monomial v -> Monomial v -> Bool

(/=) :: Monomial v -> Monomial v -> Bool

(Eq v, Show v) => Num (Monomial v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

(+) :: Monomial v -> Monomial v -> Monomial v

(-) :: Monomial v -> Monomial v -> Monomial v

(*) :: Monomial v -> Monomial v -> Monomial v

negate :: Monomial v -> Monomial v

abs :: Monomial v -> Monomial v

signum :: Monomial v -> Monomial v

fromInteger :: Integer -> Monomial v

Ord v => Ord (Monomial v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

compare :: Monomial v -> Monomial v -> Ordering

(<) :: Monomial v -> Monomial v -> Bool

(<=) :: Monomial v -> Monomial v -> Bool

(>) :: Monomial v -> Monomial v -> Bool

(>=) :: Monomial v -> Monomial v -> Bool

max :: Monomial v -> Monomial v -> Monomial v

min :: Monomial v -> Monomial v -> Monomial v

(Eq v, Show v) => Show (Monomial v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

showsPrec :: Int -> Monomial v -> ShowS

show :: Monomial v -> String

showList :: [Monomial v] -> ShowS

divM :: Eq v => Monomial v -> Monomial v -> Maybe (Monomial v, Monomial v) Source #

newtype NPoly r v Source #

Constructors

NP [(Monomial v, r)] 

Instances

Instances details
(Eq v, Eq r) => Eq (NPoly r v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

(==) :: NPoly r v -> NPoly r v -> Bool

(/=) :: NPoly r v -> NPoly r v -> Bool

(Eq k, Fractional k, Ord v, Show v) => Fractional (NPoly k v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

(/) :: NPoly k v -> NPoly k v -> NPoly k v

recip :: NPoly k v -> NPoly k v

fromRational :: Rational -> NPoly k v

(Eq r, Num r, Ord v, Show v) => Num (NPoly r v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

(+) :: NPoly r v -> NPoly r v -> NPoly r v

(-) :: NPoly r v -> NPoly r v -> NPoly r v

(*) :: NPoly r v -> NPoly r v -> NPoly r v

negate :: NPoly r v -> NPoly r v

abs :: NPoly r v -> NPoly r v

signum :: NPoly r v -> NPoly r v

fromInteger :: Integer -> NPoly r v

(Ord r, Ord v) => Ord (NPoly r v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

compare :: NPoly r v -> NPoly r v -> Ordering

(<) :: NPoly r v -> NPoly r v -> Bool

(<=) :: NPoly r v -> NPoly r v -> Bool

(>) :: NPoly r v -> NPoly r v -> Bool

(>=) :: NPoly r v -> NPoly r v -> Bool

max :: NPoly r v -> NPoly r v -> NPoly r v

min :: NPoly r v -> NPoly r v -> NPoly r v

(Show r, Eq v, Show v) => Show (NPoly r v) Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

showsPrec :: Int -> NPoly r v -> ShowS

show :: NPoly r v -> String

showList :: [NPoly r v] -> ShowS

Invertible (NPoly LPQ BraidGens) Source # 
Instance details

Defined in Math.Projects.KnotTheory.Braid

Invertible (NPoly LPQ IwahoriHeckeGens) Source # 
Instance details

Defined in Math.Projects.KnotTheory.IwahoriHecke

cmpTerm :: Ord a => (a, b1) -> (a, b2) -> Ordering Source #

mergeTerms :: (Ord a, Eq b, Num b) => [(a, b)] -> [(a, b)] -> [(a, b)] Source #

collect :: (Num a1, Eq a2, Eq a1) => [(a2, a1)] -> [(a2, a1)] Source #

data Var Source #

Constructors

X 
Y 
Z 

Instances

Instances details
Eq Var Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

(==) :: Var -> Var -> Bool

(/=) :: Var -> Var -> Bool

Ord Var Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

compare :: Var -> Var -> Ordering

(<) :: Var -> Var -> Bool

(<=) :: Var -> Var -> Bool

(>) :: Var -> Var -> Bool

(>=) :: Var -> Var -> Bool

max :: Var -> Var -> Var

min :: Var -> Var -> Var

Show Var Source # 
Instance details

Defined in Math.Algebra.NonCommutative.NCPoly

Methods

showsPrec :: Int -> Var -> ShowS

show :: Var -> String

showList :: [Var] -> ShowS

var :: Num k => v -> NPoly k v Source #

Create a non-commutative variable for use in forming non-commutative polynomials. For example, we could define x = var "x", y = var "y". Then x*y /= y*x.

lm :: NPoly r v -> Monomial v Source #

lc :: NPoly r v -> r Source #

lt :: NPoly r v -> NPoly r v Source #

quotRemNP :: forall v r. (Fractional r, Ord v, Show v, Eq r) => NPoly r v -> [NPoly r v] -> ([(NPoly r v, NPoly r v)], NPoly r v) Source #

remNP :: forall v r. (Fractional r, Ord v, Show v, Eq r) => NPoly r v -> [NPoly r v] -> NPoly r v Source #

(%%) :: (Fractional r, Ord v, Show v, Eq r) => NPoly r v -> [NPoly r v] -> NPoly r v infixl 7 Source #

remNP2 :: forall v r. (Num r, Ord v, Show v, Eq r) => NPoly r v -> [NPoly r v] -> NPoly r v Source #

toMonic :: forall v r. (Eq r, Ord v, Show v, Fractional r) => NPoly r v -> NPoly r v Source #

inject :: (Num r, Eq r, Eq v, Show v) => r -> NPoly r v Source #

subst :: (Num r1, Ord v1, Show v1, Eq r1, Eq v2, Eq r2, Show r2, Show v2, Num r2) => [(NPoly r2 v2, NPoly r1 v1)] -> NPoly r1 v2 -> NPoly r1 v1 Source #

class Invertible a where Source #

Methods

inv :: a -> a Source #

(^-) :: (Integral b, Invertible a, Num a) => a -> b -> a Source #