(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule Integer (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*.) :: OrderedPolynomial r order n -> Integer -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule Natural (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*.) :: OrderedPolynomial r order n -> Natural -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule Integer (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (.*) :: Integer -> OrderedPolynomial r order n -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule Natural (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (.*) :: Natural -> OrderedPolynomial r order n -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule (Scalar r) (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*.) :: OrderedPolynomial r order n -> Scalar r -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule (Scalar r) (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (.*) :: Scalar r -> OrderedPolynomial r order n -> OrderedPolynomial r order n #
Eq r => Eq (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (==) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> Bool # (/=) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> Bool #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Num (OrderedPolynomial r order n) Source #	We provide Num instance to use trivial injection R into R[X]. Do not use signum or abs.
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (+) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # (-) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # (*) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # negate :: OrderedPolynomial r order n -> OrderedPolynomial r order n # abs :: OrderedPolynomial r order n -> OrderedPolynomial r order n # signum :: OrderedPolynomial r order n -> OrderedPolynomial r order n # fromInteger :: Integer -> OrderedPolynomial r order n #
(CoeffRing r, IsOrder n ord, Ord r) => Ord (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods compare :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Ordering # (<) :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Bool # (<=) :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Bool # (>) :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Bool # (>=) :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Bool # max :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # min :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n #
(KnownNat n, CoeffRing r, IsMonomialOrder n order, PrettyCoeff r) => Show (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods showsPrec :: Int -> OrderedPolynomial r order n -> ShowS # show :: OrderedPolynomial r order n -> String # showList :: [OrderedPolynomial r order n] -> ShowS #
(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => UFD (OrderedPolynomial r ord 1) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(KnownNat n, Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => UFD (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial
(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => PID (OrderedPolynomial r ord 1) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods egcd :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> (OrderedPolynomial r ord 1, OrderedPolynomial r ord 1, OrderedPolynomial r ord 1) #
(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => GCDDomain (OrderedPolynomial r ord 1) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods gcd :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 # reduceFraction :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> (OrderedPolynomial r ord 1, OrderedPolynomial r ord 1) # lcm :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 #
(KnownNat n, Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => GCDDomain (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial Methods gcd :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # reduceFraction :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> (OrderedPolynomial r ord n, OrderedPolynomial r ord n) # lcm :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n #
(Eq r, DecidableUnits r, DecidableZero r, KnownNat n, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => IntegralDomain (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods divides :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Bool # maybeQuot :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Maybe (OrderedPolynomial r ord n) #
(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => IntegralDomain (OrderedPolynomial r ord 1) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods divides :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> Bool # maybeQuot :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> Maybe (OrderedPolynomial r ord 1) #
(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => Euclidean (OrderedPolynomial r ord 1) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods degree :: OrderedPolynomial r ord 1 -> Maybe Natural # divide :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> (OrderedPolynomial r ord 1, OrderedPolynomial r ord 1) # quot :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 # rem :: OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 -> OrderedPolynomial r ord 1 #
(Eq r, DecidableUnits r, DecidableZero r, KnownNat n, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => ZeroProductSemiring (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(Eq r, Euclidean r, KnownNat n, IsMonomialOrder n ord) => UnitNormalForm (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods splitUnit :: OrderedPolynomial r ord n -> (OrderedPolynomial r ord n, OrderedPolynomial r ord n) #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Ring (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods fromInteger :: Integer -> OrderedPolynomial r order n
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Rig (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods fromNatural :: Natural -> OrderedPolynomial r order n #
(IsMonomialOrder n ord, Characteristic r, KnownNat n, CoeffRing r) => Characteristic (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods char :: proxy (OrderedPolynomial r ord n) -> Natural #
(CoeffRing r, KnownNat n, IsMonomialOrder n ord) => DecidableZero (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods isZero :: OrderedPolynomial r ord n -> Bool #
(CoeffRing r, IsMonomialOrder n ord, DecidableUnits r, KnownNat n) => DecidableUnits (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods recipUnit :: OrderedPolynomial r ord n -> Maybe (OrderedPolynomial r ord n) # isUnit :: OrderedPolynomial r ord n -> Bool # (^?) :: Integral n0 => OrderedPolynomial r ord n -> n0 -> Maybe (OrderedPolynomial r ord n) #
(Eq r, KnownNat n, Euclidean r, IsMonomialOrder n ord) => DecidableAssociates (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods isAssociate :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> Bool #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Unital (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods one :: OrderedPolynomial r order n # pow :: OrderedPolynomial r order n -> Natural -> OrderedPolynomial r order n # productWith :: Foldable f => (a -> OrderedPolynomial r order n) -> f a -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Commutative (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Semiring (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Multiplicative (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # pow1p :: OrderedPolynomial r order n -> Natural -> OrderedPolynomial r order n # productWith1 :: Foldable1 f => (a -> OrderedPolynomial r order n) -> f a -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Monoidal (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods zero :: OrderedPolynomial r order n # sinnum :: Natural -> OrderedPolynomial r order n -> OrderedPolynomial r order n # sumWith :: Foldable f => (a -> OrderedPolynomial r order n) -> f a -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Group (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (-) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # negate :: OrderedPolynomial r order n -> OrderedPolynomial r order n # subtract :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # times :: Integral n0 => n0 -> OrderedPolynomial r order n -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Additive (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (+) :: OrderedPolynomial r order n -> OrderedPolynomial r order n -> OrderedPolynomial r order n # sinnum1p :: Natural -> OrderedPolynomial r order n -> OrderedPolynomial r order n # sumWith1 :: Foldable1 f => (a -> OrderedPolynomial r order n) -> f a -> OrderedPolynomial r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Abelian (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
Hashable r => Hashable (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods hashWithSalt :: Int -> OrderedPolynomial r ord n -> Int # hash :: OrderedPolynomial r ord n -> Int #
NFData r => NFData (OrderedPolynomial r order n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods rnf :: OrderedPolynomial r order n -> () #
(KnownNat n, IsMonomialOrder n ord, CoeffRing r) => IsPolynomial (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Associated Types type Coefficient (OrderedPolynomial r ord n) # type Arity (OrderedPolynomial r ord n) :: Nat # Methods liftMap :: (Module (Scalar (Coefficient (OrderedPolynomial r ord n))) alg, Ring alg, Commutative alg) => (Ordinal (Arity (OrderedPolynomial r ord n)) -> alg) -> OrderedPolynomial r ord n -> alg # subst :: (Ring alg, Commutative alg, Module (Scalar (Coefficient (OrderedPolynomial r ord n))) alg) => Sized (Arity (OrderedPolynomial r ord n)) alg -> OrderedPolynomial r ord n -> alg # substWith :: Ring m => (Coefficient (OrderedPolynomial r ord n) -> m -> m) -> Sized (Arity (OrderedPolynomial r ord n)) m -> OrderedPolynomial r ord n -> m # sArity' :: OrderedPolynomial r ord n -> SNat (Arity (OrderedPolynomial r ord n)) # sArity :: proxy (OrderedPolynomial r ord n) -> SNat (Arity (OrderedPolynomial r ord n)) # arity :: proxy (OrderedPolynomial r ord n) -> Integer # injectCoeff :: Coefficient (OrderedPolynomial r ord n) -> OrderedPolynomial r ord n # injectCoeff' :: proxy (OrderedPolynomial r ord n) -> Coefficient (OrderedPolynomial r ord n) -> OrderedPolynomial r ord n # monomials :: OrderedPolynomial r ord n -> HashSet (Monomial (Arity (OrderedPolynomial r ord n))) # terms' :: OrderedPolynomial r ord n -> Map (Monomial (Arity (OrderedPolynomial r ord n))) (Coefficient (OrderedPolynomial r ord n)) # coeff' :: Monomial (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n -> Coefficient (OrderedPolynomial r ord n) # constantTerm :: OrderedPolynomial r ord n -> Coefficient (OrderedPolynomial r ord n) # fromMonomial :: Monomial (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # toPolynomial' :: (Coefficient (OrderedPolynomial r ord n), Monomial (Arity (OrderedPolynomial r ord n))) -> OrderedPolynomial r ord n # polynomial' :: Map (Monomial (Arity (OrderedPolynomial r ord n))) (Coefficient (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # totalDegree' :: OrderedPolynomial r ord n -> Int # var :: Ordinal (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # mapCoeff' :: (Coefficient (OrderedPolynomial r ord n) -> Coefficient (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # (>\|) :: Monomial (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # (\|<) :: OrderedPolynomial r ord n -> Monomial (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # (!*) :: Coefficient (OrderedPolynomial r ord n) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # _Terms' :: Iso' (OrderedPolynomial r ord n) (Map (Monomial (Arity (OrderedPolynomial r ord n))) (Coefficient (OrderedPolynomial r ord n))) # mapMonomial :: (Monomial (Arity (OrderedPolynomial r ord n)) -> Monomial (Arity (OrderedPolynomial r ord n))) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n #
(KnownNat n, CoeffRing r, IsMonomialOrder n ord) => IsOrderedPolynomial (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Associated Types type MOrder (OrderedPolynomial r ord n) # Methods coeff :: OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n -> Coefficient (OrderedPolynomial r ord n) # terms :: OrderedPolynomial r ord n -> Map (OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) (Coefficient (OrderedPolynomial r ord n)) # leadingTerm :: OrderedPolynomial r ord n -> (Coefficient (OrderedPolynomial r ord n), OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) # leadingMonomial :: OrderedPolynomial r ord n -> OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n)) # leadingCoeff :: OrderedPolynomial r ord n -> Coefficient (OrderedPolynomial r ord n) # splitLeadingTerm :: OrderedPolynomial r ord n -> (Term (OrderedPolynomial r ord n), OrderedPolynomial r ord n) # orderedMonomials :: OrderedPolynomial r ord n -> Set (OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) # fromOrderedMonomial :: OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # toPolynomial :: (Coefficient (OrderedPolynomial r ord n), OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) -> OrderedPolynomial r ord n # polynomial :: Map (OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) (Coefficient (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # (>) :: OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # (<) :: OrderedPolynomial r ord n -> OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n # _Terms :: Iso' (OrderedPolynomial r ord n) (Map (OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) (Coefficient (OrderedPolynomial r ord n))) # diff :: Ordinal (Arity (OrderedPolynomial r ord n)) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # mapMonomialMonotonic :: (OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n)) -> OrderedMonomial (MOrder (OrderedPolynomial r ord n)) (Arity (OrderedPolynomial r ord n))) -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n #
type Arity (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal type Arity (OrderedPolynomial r ord n) = n
type Coefficient (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal type Coefficient (OrderedPolynomial r ord n) = r
type MOrder (OrderedPolynomial r ord n) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal type MOrder (OrderedPolynomial r ord n) = ord

allVars :: forall k ord n. (IsMonomialOrder n ord, CoeffRing k, KnownNat n) => Sized n (OrderedPolynomial k ord n) Source #

substVar :: (CoeffRing r, KnownNat n, IsMonomialOrder n ord, 1 <= n) => Ordinal n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n Source #

substVar n f substitutes n-th variable with polynomial f, without changing arity.

homogenize :: forall k ord n. (CoeffRing k, KnownNat n, IsMonomialOrder (n + 1) ord, IsMonomialOrder n ord) => OrderedPolynomial k ord n -> OrderedPolynomial k ord (n + 1) Source #

Calculate the homogenized polynomial of given one, with additional variable is the last variable.

unhomogenize :: forall k ord n. (CoeffRing k, KnownNat n, IsMonomialOrder n ord, IsMonomialOrder (n + 1) ord) => OrderedPolynomial k ord (n + 1) -> OrderedPolynomial k ord n Source #

normalize :: DecidableZero r => OrderedPolynomial r order n -> OrderedPolynomial r order n Source #

varX :: forall r n order. (CoeffRing r, KnownNat n, IsMonomialOrder n order, 0 < n) => OrderedPolynomial r order n Source #

getTerms :: OrderedPolynomial k order n -> [(k, OrderedMonomial order n)] Source #

shiftR :: forall k r n ord. (CoeffRing r, KnownNat n, IsMonomialOrder n ord, IsMonomialOrder (k + n) ord) => SNat k -> OrderedPolynomial r ord n -> OrderedPolynomial r ord (k + n) Source #

orderedBy :: OrderedPolynomial k o n -> o -> OrderedPolynomial k o n Source #

mapCoeff :: (KnownNat n, CoeffRing b, IsMonomialOrder n ord) => (a -> b) -> OrderedPolynomial a ord n -> OrderedPolynomial b ord n Source #

reversal :: (CoeffRing k, IsMonomialOrder 1 o) => Int -> OrderedPolynomial k o 1 -> OrderedPolynomial k o 1 Source #

padeApprox :: (Field r, DecidableUnits r, CoeffRing r, ZeroProductSemiring r, IsMonomialOrder 1 order) => Natural -> Natural -> OrderedPolynomial r order 1 -> (OrderedPolynomial r order 1, OrderedPolynomial r order 1) Source #

eval :: (CoeffRing r, IsMonomialOrder n order, KnownNat n) => Sized n r -> OrderedPolynomial r order n -> r Source #

Evaluate polynomial at some point.

evalUnivariate :: (CoeffRing b, IsMonomialOrder 1 order) => b -> OrderedPolynomial b order 1 -> b Source #

substUnivariate :: (Module (Scalar r) b, Unital b, CoeffRing r, IsMonomialOrder 1 order) => b -> OrderedPolynomial r order 1 -> b Source #

Substitute univariate polynomial using Horner's rule

minpolRecurrent :: forall k. (Eq k, ZeroProductSemiring k, DecidableUnits k, DecidableZero k, Field k) => Natural -> [k] -> Polynomial k 1 Source #

class IsOrder (n :: Nat) ordering where #

Class to lookup ordering from its (type-level) name.

Methods

cmpMonomial :: Proxy ordering -> MonomialOrder n #

Instances

Instances details

KnownNat n => IsOrder n Revlex
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy Revlex -> MonomialOrder n #
KnownNat n => IsOrder n Lex
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy Lex -> MonomialOrder n #
KnownNat n => IsOrder n Grlex
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy Grlex -> MonomialOrder n #
KnownNat n => IsOrder n Grevlex
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy Grevlex -> MonomialOrder n #
IsOrder n ord => IsOrder n (Graded ord)
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy (Graded ord) -> MonomialOrder n #
(KnownNat n, IsOrder n ord, SingI ws) => IsOrder n (WeightOrder ws ord)
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy (WeightOrder ws ord) -> MonomialOrder n #
(IsOrder n ord, IsOrder m ord', KnownNat m, KnownNat n, k ~ (n + m)) => IsOrder k (ProductOrder n m ord ord')
Instance details Defined in Algebra.Ring.Polynomial.Monomial Methods cmpMonomial :: Proxy (ProductOrder n m ord ord') -> MonomialOrder k #

newtype PadPolyL n ord poly Source #

Constructors

PadPolyL
Fields runPadPolyL :: OrderedPolynomial poly (Graded ord) n

Instances

Instances details

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => RightModule Integer (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*.) :: PadPolyL n ord poly -> Integer -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => RightModule Natural (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*.) :: PadPolyL n ord poly -> Natural -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => LeftModule Integer (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (.*) :: Integer -> PadPolyL n ord poly -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => LeftModule Natural (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (.*) :: Natural -> PadPolyL n ord poly -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, RightModule (Scalar r) poly) => RightModule (Scalar r) (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*.) :: PadPolyL n ord poly -> Scalar r -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, LeftModule (Scalar r) poly) => LeftModule (Scalar r) (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (.*) :: Scalar r -> PadPolyL n ord poly -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Eq (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (==) :: PadPolyL n ord poly -> PadPolyL n ord poly -> Bool # (/=) :: PadPolyL n ord poly -> PadPolyL n ord poly -> Bool #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Num (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (+) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # (-) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # (*) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # negate :: PadPolyL n ord poly -> PadPolyL n ord poly # abs :: PadPolyL n ord poly -> PadPolyL n ord poly # signum :: PadPolyL n ord poly -> PadPolyL n ord poly # fromInteger :: Integer -> PadPolyL n ord poly #
(r ~ Coefficient poly, IsOrderedPolynomial poly, SingI (Replicate n 1), KnownNat n, CoeffRing r, IsMonomialOrder n order, PrettyCoeff r) => Show (PadPolyL n order poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods showsPrec :: Int -> PadPolyL n order poly -> ShowS # show :: PadPolyL n order poly -> String # showList :: [PadPolyL n order poly] -> ShowS #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Ring (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods fromInteger :: Integer -> PadPolyL n ord poly
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Rig (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods fromNatural :: Natural -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => DecidableZero (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods isZero :: PadPolyL n ord poly -> Bool #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Unital (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods one :: PadPolyL n ord poly # pow :: PadPolyL n ord poly -> Natural -> PadPolyL n ord poly # productWith :: Foldable f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Commutative (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Semiring (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Multiplicative (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (*) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # pow1p :: PadPolyL n ord poly -> Natural -> PadPolyL n ord poly # productWith1 :: Foldable1 f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Monoidal (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods zero :: PadPolyL n ord poly # sinnum :: Natural -> PadPolyL n ord poly -> PadPolyL n ord poly # sumWith :: Foldable f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Group (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (-) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # negate :: PadPolyL n ord poly -> PadPolyL n ord poly # subtract :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # times :: Integral n0 => n0 -> PadPolyL n ord poly -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Additive (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Methods (+) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly # sinnum1p :: Natural -> PadPolyL n ord poly -> PadPolyL n ord poly # sumWith1 :: Foldable1 f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Abelian (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => IsPolynomial (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Associated Types type Coefficient (PadPolyL n ord poly) # type Arity (PadPolyL n ord poly) :: Nat # Methods liftMap :: (Module (Scalar (Coefficient (PadPolyL n ord poly))) alg, Ring alg, Commutative alg) => (Ordinal (Arity (PadPolyL n ord poly)) -> alg) -> PadPolyL n ord poly -> alg # subst :: (Ring alg, Commutative alg, Module (Scalar (Coefficient (PadPolyL n ord poly))) alg) => Sized (Arity (PadPolyL n ord poly)) alg -> PadPolyL n ord poly -> alg # substWith :: Ring m => (Coefficient (PadPolyL n ord poly) -> m -> m) -> Sized (Arity (PadPolyL n ord poly)) m -> PadPolyL n ord poly -> m # sArity' :: PadPolyL n ord poly -> SNat (Arity (PadPolyL n ord poly)) # sArity :: proxy (PadPolyL n ord poly) -> SNat (Arity (PadPolyL n ord poly)) # arity :: proxy (PadPolyL n ord poly) -> Integer # injectCoeff :: Coefficient (PadPolyL n ord poly) -> PadPolyL n ord poly # injectCoeff' :: proxy (PadPolyL n ord poly) -> Coefficient (PadPolyL n ord poly) -> PadPolyL n ord poly # monomials :: PadPolyL n ord poly -> HashSet (Monomial (Arity (PadPolyL n ord poly))) # terms' :: PadPolyL n ord poly -> Map (Monomial (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)) # coeff' :: Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly) # constantTerm :: PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly) # fromMonomial :: Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly # toPolynomial' :: (Coefficient (PadPolyL n ord poly), Monomial (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly # polynomial' :: Map (Monomial (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)) -> PadPolyL n ord poly # totalDegree' :: PadPolyL n ord poly -> Int # var :: Ordinal (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly # mapCoeff' :: (Coefficient (PadPolyL n ord poly) -> Coefficient (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly # (>\|) :: Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly # (\|<) :: PadPolyL n ord poly -> Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly # (!*) :: Coefficient (PadPolyL n ord poly) -> PadPolyL n ord poly -> PadPolyL n ord poly # _Terms' :: Iso' (PadPolyL n ord poly) (Map (Monomial (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly))) # mapMonomial :: (Monomial (Arity (PadPolyL n ord poly)) -> Monomial (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly -> PadPolyL n ord poly #
(SingI (Replicate n 1), KnownNat n, IsMonomialOrder n ord, IsOrderedPolynomial poly) => IsOrderedPolynomial (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal Associated Types type MOrder (PadPolyL n ord poly) # Methods coeff :: OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly) # terms :: PadPolyL n ord poly -> Map (OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)) # leadingTerm :: PadPolyL n ord poly -> (Coefficient (PadPolyL n ord poly), OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) # leadingMonomial :: PadPolyL n ord poly -> OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) # leadingCoeff :: PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly) # splitLeadingTerm :: PadPolyL n ord poly -> (Term (PadPolyL n ord poly), PadPolyL n ord poly) # orderedMonomials :: PadPolyL n ord poly -> Set (OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) # fromOrderedMonomial :: OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly # toPolynomial :: (Coefficient (PadPolyL n ord poly), OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly # polynomial :: Map (OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)) -> PadPolyL n ord poly # (>) :: OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly # (<) :: PadPolyL n ord poly -> OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly # _Terms :: Iso' (PadPolyL n ord poly) (Map (OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly))) # diff :: Ordinal (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly # mapMonomialMonotonic :: (OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> OrderedMonomial (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly -> PadPolyL n ord poly #
type Arity (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal type Arity (PadPolyL n ord poly) = n + Arity poly
type Coefficient (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal type Coefficient (PadPolyL n ord poly) = Coefficient poly
type MOrder (PadPolyL n ord poly) Source #
Instance details Defined in Algebra.Ring.Polynomial.Internal type MOrder (PadPolyL n ord poly) = ProductOrder n (Arity poly) (Graded ord) (MOrder poly)

padLeftPoly :: (IsMonomialOrder n ord, IsPolynomial poly) => SNat n -> ord -> poly -> PadPolyL n ord poly Source #

Orphan instances

(KnownNat n, Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => UFD (OrderedPolynomial r ord n) Source #
Instance details
(KnownNat n, Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => GCDDomain (OrderedPolynomial r ord n) Source #
Instance details Methods gcd :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n # reduceFraction :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> (OrderedPolynomial r ord n, OrderedPolynomial r ord n) # lcm :: OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n #