halg-algorithms-0.6.0.0: Algorithms related to Gröbner basis, part of halg computational algebra suite.
Safe HaskellNone
LanguageHaskell2010

Algebra.Field.RationalFunction

Synopsis

Documentation

data RationalFunction k Source #

Unary rational field over a field k.

With OverloadedLabels extension, you can use IsLabel instance to write variable as #x; for example 1 / (#x - 2) ^ n.

Instances

Instances details
(Field k, Eq k) => IsLabel "x" (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

fromLabel :: RationalFunction k #

(Eq k, Euclidean k, Division k) => RightModule Integer (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(*.) :: RationalFunction k -> Integer -> RationalFunction k #

(Eq k, Euclidean k, Division k) => RightModule Natural (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(*.) :: RationalFunction k -> Natural -> RationalFunction k #

(Eq k, Euclidean k, Division k) => LeftModule Integer (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(.*) :: Integer -> RationalFunction k -> RationalFunction k #

(Eq k, Euclidean k, Division k) => LeftModule Natural (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(.*) :: Natural -> RationalFunction k -> RationalFunction k #

(Eq k, Euclidean k, Division k) => Eq (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(==) :: RationalFunction k -> RationalFunction k -> Bool #

(/=) :: RationalFunction k -> RationalFunction k -> Bool #

(Eq k, Euclidean k, Division k) => Num (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(+) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(-) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(*) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

negate :: RationalFunction k -> RationalFunction k #

abs :: RationalFunction k -> RationalFunction k #

signum :: RationalFunction k -> RationalFunction k #

fromInteger :: Integer -> RationalFunction k #

(Euclidean k, Division k, Ord k) => Ord (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

compare :: RationalFunction k -> RationalFunction k -> Ordering #

(<) :: RationalFunction k -> RationalFunction k -> Bool #

(<=) :: RationalFunction k -> RationalFunction k -> Bool #

(>) :: RationalFunction k -> RationalFunction k -> Bool #

(>=) :: RationalFunction k -> RationalFunction k -> Bool #

max :: RationalFunction k -> RationalFunction k -> RationalFunction k #

min :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(CoeffRing k, PrettyCoeff k) => Show (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

showsPrec :: Int -> RationalFunction k -> ShowS #

show :: RationalFunction k -> String #

showList :: [RationalFunction k] -> ShowS #

(Eq k, Euclidean k, Division k) => UFD (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

(Eq k, Euclidean k, Division k) => PID (RationalFunction k) Source # 
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Defined in Algebra.Field.RationalFunction

Methods

egcd :: RationalFunction k -> RationalFunction k -> (RationalFunction k, RationalFunction k, RationalFunction k) #

(Eq k, Euclidean k, Division k) => GCDDomain (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

gcd :: RationalFunction k -> RationalFunction k -> RationalFunction k #

reduceFraction :: RationalFunction k -> RationalFunction k -> (RationalFunction k, RationalFunction k) #

lcm :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(Eq k, Euclidean k, Division k) => IntegralDomain (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

divides :: RationalFunction k -> RationalFunction k -> Bool #

maybeQuot :: RationalFunction k -> RationalFunction k -> Maybe (RationalFunction k) #

(Eq k, Euclidean k, Division k) => Euclidean (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

degree :: RationalFunction k -> Maybe Natural #

divide :: RationalFunction k -> RationalFunction k -> (RationalFunction k, RationalFunction k) #

quot :: RationalFunction k -> RationalFunction k -> RationalFunction k #

rem :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(Eq k, Euclidean k, Division k) => ZeroProductSemiring (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

(Eq k, Euclidean k, Division k) => UnitNormalForm (RationalFunction k) Source # 
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Defined in Algebra.Field.RationalFunction

Methods

splitUnit :: RationalFunction k -> (RationalFunction k, RationalFunction k) #

(Eq k, Euclidean k, Division k) => Ring (RationalFunction k) Source # 
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Defined in Algebra.Field.RationalFunction

Methods

fromInteger :: Integer -> RationalFunction k

(Eq k, Euclidean k, Division k) => Rig (RationalFunction k) Source # 
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Defined in Algebra.Field.RationalFunction

Methods

fromNatural :: Natural -> RationalFunction k #

(Eq k, Euclidean k, Division k) => DecidableZero (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

isZero :: RationalFunction k -> Bool #

(Eq k, Euclidean k, Division k) => DecidableUnits (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

recipUnit :: RationalFunction k -> Maybe (RationalFunction k) #

isUnit :: RationalFunction k -> Bool #

(^?) :: Integral n => RationalFunction k -> n -> Maybe (RationalFunction k) #

(Eq k, Euclidean k, Division k) => DecidableAssociates (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

isAssociate :: RationalFunction k -> RationalFunction k -> Bool #

(Eq k, Euclidean k, Division k) => Unital (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

one :: RationalFunction k #

pow :: RationalFunction k -> Natural -> RationalFunction k #

productWith :: Foldable f => (a -> RationalFunction k) -> f a -> RationalFunction k #

(Eq k, Euclidean k, Division k) => Division (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

recip :: RationalFunction k -> RationalFunction k #

(/) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(\\) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

(^) :: Integral n => RationalFunction k -> n -> RationalFunction k

(Eq k, Euclidean k, Division k) => Commutative (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

(Eq k, Euclidean k, Division k) => Semiring (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

(Eq k, Euclidean k, Division k) => Multiplicative (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(*) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

pow1p :: RationalFunction k -> Natural -> RationalFunction k #

productWith1 :: Foldable1 f => (a -> RationalFunction k) -> f a -> RationalFunction k #

(Eq k, Euclidean k, Division k) => Monoidal (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

zero :: RationalFunction k #

sinnum :: Natural -> RationalFunction k -> RationalFunction k #

sumWith :: Foldable f => (a -> RationalFunction k) -> f a -> RationalFunction k #

(Eq k, Euclidean k, Division k) => Group (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(-) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

negate :: RationalFunction k -> RationalFunction k #

subtract :: RationalFunction k -> RationalFunction k -> RationalFunction k #

times :: Integral n => n -> RationalFunction k -> RationalFunction k #

(Eq k, Euclidean k, Division k) => Additive (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

Methods

(+) :: RationalFunction k -> RationalFunction k -> RationalFunction k #

sinnum1p :: Natural -> RationalFunction k -> RationalFunction k #

sumWith1 :: Foldable1 f => (a -> RationalFunction k) -> f a -> RationalFunction k #

(Eq k, Euclidean k, Division k) => Abelian (RationalFunction k) Source # 
Instance details

Defined in Algebra.Field.RationalFunction

evalRationalFunctionOnField :: (Eq k, Field k) => RationalFunction k -> k -> k Source #

evalRationalFunction :: (GCDDomain r, CoeffRing r) => RationalFunction r -> r -> Fraction r Source #

fromPolynomial :: (IsOrderedPolynomial poly, Arity poly ~ 1, Field (Coefficient poly)) => poly -> RationalFunction (Coefficient poly) Source #

variable :: (Eq k, Field k) => RationalFunction k Source #

diffRF :: (Eq k, Field k) => RationalFunction k -> RationalFunction k Source #

Formal differentiation

taylor :: (Eq k, Field k) => RationalFunction k -> [k] Source #

Formal Taylor expansion

taylorCoeff :: (Eq k, Field k) => Int -> RationalFunction k -> k Source #