Safe Haskell	None
Language	Haskell2010

Algebra.Scalar

Synopsis

Documentation

newtype Scalar r Source #

Scalar r provides almost the same type-instances as r, but it can also behave as a Module over r itself.

Constructors

Scalar
Fields runScalar :: r

Instances

LeftModule Integer r => LeftModule Integer (Scalar r) Source #
Methods (.*) :: Integer -> Scalar r -> Scalar r #
Semiring r => LeftModule r (Scalar r) Source #
Methods (.*) :: r -> Scalar r -> Scalar r #
LeftModule Natural r => LeftModule Natural (Scalar r) Source #
Methods (.*) :: Natural -> Scalar r -> Scalar r #
RightModule Integer r => RightModule Integer (Scalar r) Source #
Methods (*.) :: Scalar r -> Integer -> Scalar r #
Semiring r => RightModule r (Scalar r) Source #
Methods (*.) :: Scalar r -> r -> Scalar r #
RightModule Natural r => RightModule Natural (Scalar r) Source #
Methods (*.) :: Scalar r -> Natural -> Scalar r #
Enum r => Enum (Scalar r) Source #
Methods succ :: Scalar r -> Scalar r # pred :: Scalar r -> Scalar r # toEnum :: Int -> Scalar r # fromEnum :: Scalar r -> Int # enumFrom :: Scalar r -> [Scalar r] # enumFromThen :: Scalar r -> Scalar r -> [Scalar r] # enumFromTo :: Scalar r -> Scalar r -> [Scalar r] # enumFromThenTo :: Scalar r -> Scalar r -> Scalar r -> [Scalar r] #
Eq r => Eq (Scalar r) Source #
Methods (==) :: Scalar r -> Scalar r -> Bool # (/=) :: Scalar r -> Scalar r -> Bool #
Fractional r => Fractional (Scalar r) Source #
Methods (/) :: Scalar r -> Scalar r -> Scalar r # recip :: Scalar r -> Scalar r # fromRational :: Rational -> Scalar r #
Integral r => Integral (Scalar r) Source #
Methods quot :: Scalar r -> Scalar r -> Scalar r # rem :: Scalar r -> Scalar r -> Scalar r # div :: Scalar r -> Scalar r -> Scalar r # mod :: Scalar r -> Scalar r -> Scalar r # quotRem :: Scalar r -> Scalar r -> (Scalar r, Scalar r) # divMod :: Scalar r -> Scalar r -> (Scalar r, Scalar r) # toInteger :: Scalar r -> Integer #
Num r => Num (Scalar r) Source #
Methods (+) :: Scalar r -> Scalar r -> Scalar r # (-) :: Scalar r -> Scalar r -> Scalar r # (*) :: Scalar r -> Scalar r -> Scalar r # negate :: Scalar r -> Scalar r # abs :: Scalar r -> Scalar r # signum :: Scalar r -> Scalar r # fromInteger :: Integer -> Scalar r #
Ord r => Ord (Scalar r) Source #
Methods compare :: Scalar r -> Scalar r -> Ordering # (<) :: Scalar r -> Scalar r -> Bool # (<=) :: Scalar r -> Scalar r -> Bool # (>) :: Scalar r -> Scalar r -> Bool # (>=) :: Scalar r -> Scalar r -> Bool # max :: Scalar r -> Scalar r -> Scalar r # min :: Scalar r -> Scalar r -> Scalar r #
Read r => Read (Scalar r) Source #
Methods readsPrec :: Int -> ReadS (Scalar r) # readList :: ReadS [Scalar r] # readPrec :: ReadPrec (Scalar r) # readListPrec :: ReadPrec [Scalar r] #
Real r => Real (Scalar r) Source #
Methods toRational :: Scalar r -> Rational #
Show r => Show (Scalar r) Source #
Methods showsPrec :: Int -> Scalar r -> ShowS # show :: Scalar r -> String # showList :: [Scalar r] -> ShowS #
Commutative r => Commutative (Scalar r) Source #

UnitNormalForm r => UnitNormalForm (Scalar r) Source #
Methods splitUnit :: Scalar r -> (Scalar r, Scalar r) #
Ring r => Ring (Scalar r) Source #
Methods fromInteger :: Integer -> Scalar r #
Rig r => Rig (Scalar r) Source #
Methods fromNatural :: Natural -> Scalar r #
DecidableUnits r => DecidableUnits (Scalar r) Source #
Methods recipUnit :: Scalar r -> Maybe (Scalar r) # isUnit :: Scalar r -> Bool # (^?) :: Integral n => Scalar r -> n -> Maybe (Scalar r) #
DecidableAssociates r => DecidableAssociates (Scalar r) Source #
Methods isAssociate :: Scalar r -> Scalar r -> Bool #
Division r => Division (Scalar r) Source #
Methods recip :: Scalar r -> Scalar r # (/) :: Scalar r -> Scalar r -> Scalar r # (\\) :: Scalar r -> Scalar r -> Scalar r # (^) :: Integral n => Scalar r -> n -> Scalar r #
Unital r => Unital (Scalar r) Source #
Methods one :: Scalar r # pow :: Scalar r -> Natural -> Scalar r # productWith :: Foldable f => (a -> Scalar r) -> f a -> Scalar r #
Group r => Group (Scalar r) Source #
Methods (-) :: Scalar r -> Scalar r -> Scalar r # negate :: Scalar r -> Scalar r # subtract :: Scalar r -> Scalar r -> Scalar r # times :: Integral n => n -> Scalar r -> Scalar r #
Multiplicative r => Multiplicative (Scalar r) Source #
Methods (*) :: Scalar r -> Scalar r -> Scalar r # pow1p :: Scalar r -> Natural -> Scalar r # productWith1 :: Foldable1 f => (a -> Scalar r) -> f a -> Scalar r #
Semiring r => Semiring (Scalar r) Source #

Monoidal r => Monoidal (Scalar r) Source #
Methods zero :: Scalar r # sinnum :: Natural -> Scalar r -> Scalar r # sumWith :: Foldable f => (a -> Scalar r) -> f a -> Scalar r #
Additive r => Additive (Scalar r) Source #
Methods (+) :: Scalar r -> Scalar r -> Scalar r # sinnum1p :: Natural -> Scalar r -> Scalar r # sumWith1 :: Foldable1 f => (a -> Scalar r) -> f a -> Scalar r #
Abelian r => Abelian (Scalar r) Source #

Normed r => Normed (Scalar r) Source #
Associated Types type Norm (Scalar r) :: * Source # Methods norm :: Scalar r -> Norm (Scalar r) Source # liftNorm :: Norm (Scalar r) -> Scalar r Source #
LeftModule (Scalar (Fraction Integer)) Algebraic #
Methods (.*) :: Scalar (Fraction Integer) -> Algebraic -> Algebraic #
RightModule (Scalar (Fraction Integer)) Algebraic #
Methods (*.) :: Algebraic -> Scalar (Fraction Integer) -> Algebraic #
Semiring r => LeftModule (Scalar r) (Scalar r) Source #
Methods (.*) :: Scalar r -> Scalar r -> Scalar r #
(DecidableZero r, Semiring r) => LeftModule (Scalar r) (Unipol r) #
Methods (.*) :: Scalar r -> Unipol r -> Unipol r #
(DecidableZero r, Semiring r, Multiplicative r) => LeftModule (Scalar r) (Matrix r) #
Methods (.*) :: Scalar r -> Matrix r -> Matrix r #
Semiring r => RightModule (Scalar r) (Scalar r) Source #
Methods (*.) :: Scalar r -> Scalar r -> Scalar r #
(DecidableZero r, Semiring r) => RightModule (Scalar r) (Unipol r) #
Methods (*.) :: Unipol r -> Scalar r -> Unipol r #
(DecidableZero r, Semiring r, Multiplicative r) => RightModule (Scalar r) (Matrix r) #
Methods (*.) :: Matrix r -> Scalar r -> Matrix r #
(Wraps vars poly, LeftModule (Scalar r) poly) => LeftModule (Scalar r) (LabPolynomial poly vars) #
Methods (.*) :: Scalar r -> LabPolynomial poly vars -> LabPolynomial poly vars #
(Wraps vars poly, RightModule (Scalar r) poly) => RightModule (Scalar r) (LabPolynomial poly vars) #
Methods (*.) :: LabPolynomial poly vars -> Scalar r -> LabPolynomial poly vars #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, LeftModule (Scalar r) poly) => LeftModule (Scalar r) (PadPolyL n ord poly) #
Methods (.*) :: Scalar r -> PadPolyL n ord poly -> PadPolyL n ord poly #
((~) * r (Coefficient poly), Field (Coefficient poly), IsOrderedPolynomial poly) => LeftModule (Scalar r) (Quotient k poly ideal) #
Methods (.*) :: Scalar r -> Quotient k poly ideal -> Quotient k poly ideal #
(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, RightModule (Scalar r) poly) => RightModule (Scalar r) (PadPolyL n ord poly) #
Methods (*.) :: PadPolyL n ord poly -> Scalar r -> PadPolyL n ord poly #
((~) * r (Coefficient poly), IsOrderedPolynomial poly) => RightModule (Scalar r) (Quotient k poly ideal) #
Methods (*.) :: Quotient k poly ideal -> Scalar r -> Quotient k poly ideal #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule (Scalar r) (OrderedPolynomial * r order n) #
Methods (.) :: Scalar r -> OrderedPolynomial r order n -> OrderedPolynomial * r order n #
(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule (Scalar r) (OrderedPolynomial * r order n) #
Methods (.) :: OrderedPolynomial r order n -> Scalar r -> OrderedPolynomial * r order n #
type Norm (Scalar r) Source #
type Norm (Scalar r) = Norm r

(.*.) :: Module (Scalar r) m => r -> m -> m infixr 8 Source #