Safe Haskell	None
Language	Haskell2010

Algebra.Ring.Polynomial.Quotient

Synopsis

data Quotient poly ideal
data QIdeal poly
reifyQuotient :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => Ideal poly -> (forall (ideal :: Type). Reifies ideal (QIdeal poly) => Proxy ideal -> a) -> a
modIdeal :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => poly -> Quotient poly ideal
modIdeal' :: (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy ideal -> poly -> Quotient poly ideal
quotRepr :: Quotient poly ideal -> poly
withQuotient :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => Ideal poly -> (forall (ideal :: Type). Reifies ideal (QIdeal poly) => Quotient poly ideal) -> poly
vectorRep :: forall poly ideal. (IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Quotient poly ideal -> Vector (Coefficient poly)
genQuotVars :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => [Quotient poly ideal]
genQuotVars' :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy ideal -> [Quotient poly ideal]
gBasis' :: Reifies ideal (QIdeal poly) => Proxy ideal -> [poly]
matRep0 :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy poly -> Proxy ideal -> OrderedMonomial (MOrder poly) (Arity poly) -> Matrix (Coefficient poly)
standardMonomials :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Maybe [Quotient poly ideal]
standardMonomials' :: (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy ideal -> Maybe [Quotient poly ideal]
matRepr' :: forall poly ideal. (Field (Coefficient poly), Reifies ideal (QIdeal poly), IsOrderedPolynomial poly) => Quotient poly ideal -> Matrix (Coefficient poly)
reduce :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => poly -> Ideal poly -> poly
multWithTable :: (IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal
multUnamb :: (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal
isZeroDimensional :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => [poly] -> Bool

Documentation

data Quotient poly ideal Source #

The polynomial modulo the ideal indexed at the last type-parameter.

Instances

Instances details

Group poly => RightModule Integer (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (*.) :: Quotient poly ideal -> Integer -> Quotient poly ideal #
Monoidal poly => RightModule Natural (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (*.) :: Quotient poly ideal -> Natural -> Quotient poly ideal #
Group poly => LeftModule Integer (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (.*) :: Integer -> Quotient poly ideal -> Quotient poly ideal #
Monoidal poly => LeftModule Natural (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (.*) :: Natural -> Quotient poly ideal -> Quotient poly ideal #
(r ~ Coefficient poly, IsOrderedPolynomial poly) => RightModule (Scalar r) (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (*.) :: Quotient poly ideal -> Scalar r -> Quotient poly ideal #
(r ~ Coefficient poly, Field (Coefficient poly), IsOrderedPolynomial poly) => LeftModule (Scalar r) (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (.*) :: Scalar r -> Quotient poly ideal -> Quotient poly ideal #
Eq poly => Eq (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (==) :: Quotient poly ideal -> Quotient poly ideal -> Bool # (/=) :: Quotient poly ideal -> Quotient poly ideal -> Bool #
(Field (Coefficient poly), UnitNormalForm poly, IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Num (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (+) :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # (-) :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # (*) :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # negate :: Quotient poly ideal -> Quotient poly ideal # abs :: Quotient poly ideal -> Quotient poly ideal # signum :: Quotient poly ideal -> Quotient poly ideal # fromInteger :: Integer -> Quotient poly ideal #
Show poly => Show (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods showsPrec :: Int -> Quotient poly ideal -> ShowS # show :: Quotient poly ideal -> String # showList :: [Quotient poly ideal] -> ShowS #
(Field (Coefficient poly), IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Ring (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods fromInteger :: Integer -> Quotient poly ideal
(Field (Coefficient poly), IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Rig (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods fromNatural :: Natural -> Quotient poly ideal #
(Field (Coefficient poly), IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Unital (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods one :: Quotient poly ideal # pow :: Quotient poly ideal -> Natural -> Quotient poly ideal # productWith :: Foldable f => (a -> Quotient poly ideal) -> f a -> Quotient poly ideal #
(Field (Coefficient poly), IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Semiring (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient
(Field (Coefficient poly), IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Multiplicative (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (*) :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # pow1p :: Quotient poly ideal -> Natural -> Quotient poly ideal # productWith1 :: Foldable1 f => (a -> Quotient poly ideal) -> f a -> Quotient poly ideal #
Monoidal poly => Monoidal (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods zero :: Quotient poly ideal # sinnum :: Natural -> Quotient poly ideal -> Quotient poly ideal # sumWith :: Foldable f => (a -> Quotient poly ideal) -> f a -> Quotient poly ideal #
Group poly => Group (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (-) :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # negate :: Quotient poly ideal -> Quotient poly ideal # subtract :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # times :: Integral n => n -> Quotient poly ideal -> Quotient poly ideal #
Additive poly => Additive (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods (+) :: Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal # sinnum1p :: Natural -> Quotient poly ideal -> Quotient poly ideal # sumWith1 :: Foldable1 f => (a -> Quotient poly ideal) -> f a -> Quotient poly ideal #
Abelian poly => Abelian (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient
NFData poly => NFData (Quotient poly ideal) Source #
Instance details Defined in Algebra.Ring.Polynomial.Quotient Methods rnf :: Quotient poly ideal -> () #

data QIdeal poly Source #

reifyQuotient :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => Ideal poly -> (forall (ideal :: Type). Reifies ideal (QIdeal poly) => Proxy ideal -> a) -> a Source #

Reifies the ideal at the type-level. The ideal can be recovered with reflect.

modIdeal :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => poly -> Quotient poly ideal Source #

Polynomial modulo ideal.

modIdeal' :: (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy ideal -> poly -> Quotient poly ideal Source #

Polynomial modulo ideal given by Proxy.

quotRepr :: Quotient poly ideal -> poly Source #

Representative polynomial of given quotient polynomial.

withQuotient :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => Ideal poly -> (forall (ideal :: Type). Reifies ideal (QIdeal poly) => Quotient poly ideal) -> poly Source #

Computes polynomial modulo ideal.

vectorRep :: forall poly ideal. (IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Quotient poly ideal -> Vector (Coefficient poly) Source #

genQuotVars :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => [Quotient poly ideal] Source #

genQuotVars' :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy ideal -> [Quotient poly ideal] Source #

gBasis' :: Reifies ideal (QIdeal poly) => Proxy ideal -> [poly] Source #

matRep0 :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy poly -> Proxy ideal -> OrderedMonomial (MOrder poly) (Arity poly) -> Matrix (Coefficient poly) Source #

standardMonomials :: forall poly ideal. (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Maybe [Quotient poly ideal] Source #

standardMonomials' :: (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Proxy ideal -> Maybe [Quotient poly ideal] Source #

Find the standard monomials of the quotient ring for the zero-dimensional ideal, which are form the basis of it as k-vector space.

matRepr' :: forall poly ideal. (Field (Coefficient poly), Reifies ideal (QIdeal poly), IsOrderedPolynomial poly) => Quotient poly ideal -> Matrix (Coefficient poly) Source #

reduce :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => poly -> Ideal poly -> poly Source #

Reduce polynomial modulo ideal.

multWithTable :: (IsOrderedPolynomial poly, Reifies ideal (QIdeal poly)) => Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal Source #

multUnamb :: (IsOrderedPolynomial poly, Field (Coefficient poly), Reifies ideal (QIdeal poly)) => Quotient poly ideal -> Quotient poly ideal -> Quotient poly ideal Source #

isZeroDimensional :: (IsOrderedPolynomial poly, Field (Coefficient poly)) => [poly] -> Bool Source #