module NumHask.Algebra.Multiplicative (
MultiplicativeMagma(..)
, MultiplicativeUnital(..)
, MultiplicativeAssociative
, MultiplicativeCommutative
, MultiplicativeInvertible(..)
, MultiplicativeHomomorphic(..)
, MultiplicativeMonoidal
, Multiplicative(..)
, MultiplicativeRightCancellative(..)
, MultiplicativeLeftCancellative(..)
, MultiplicativeGroup(..)
) where
import qualified Protolude as P
import Protolude (Double, Float, Int, Integer, Bool(..))
import Data.Complex (Complex(..))
import NumHask.Algebra.Additive
class MultiplicativeMagma a where times :: a -> a -> a
instance MultiplicativeMagma Double where times = (P.*)
instance MultiplicativeMagma Float where times = (P.*)
instance MultiplicativeMagma Int where times = (P.*)
instance MultiplicativeMagma Integer where times = (P.*)
instance MultiplicativeMagma Bool where times = (P.&&)
instance (MultiplicativeMagma a, AdditiveGroup a) =>
MultiplicativeMagma (Complex a) where
(rx :+ ix) `times` (ry :+ iy) =
(rx `times` ry ix `times` iy) :+ (ix `times` ry + iy `times` rx)
class MultiplicativeMagma a => MultiplicativeUnital a where one :: a
instance MultiplicativeUnital Double where one = 1
instance MultiplicativeUnital Float where one = 1
instance MultiplicativeUnital Int where one = 1
instance MultiplicativeUnital Integer where one = 1
instance MultiplicativeUnital Bool where one = True
instance (AdditiveUnital a, AdditiveGroup a, MultiplicativeUnital a) =>
MultiplicativeUnital (Complex a) where
one = one :+ zero
class MultiplicativeMagma a => MultiplicativeAssociative a
instance MultiplicativeAssociative Double
instance MultiplicativeAssociative Float
instance MultiplicativeAssociative Int
instance MultiplicativeAssociative Integer
instance MultiplicativeAssociative Bool
instance (AdditiveGroup a, MultiplicativeAssociative a) =>
MultiplicativeAssociative (Complex a)
class MultiplicativeMagma a => MultiplicativeCommutative a
instance MultiplicativeCommutative Double
instance MultiplicativeCommutative Float
instance MultiplicativeCommutative Int
instance MultiplicativeCommutative Integer
instance MultiplicativeCommutative Bool
instance (AdditiveGroup a, MultiplicativeCommutative a) =>
MultiplicativeCommutative (Complex a)
class MultiplicativeMagma a => MultiplicativeInvertible a where recip :: a -> a
instance MultiplicativeInvertible Double where recip = P.recip
instance MultiplicativeInvertible Float where recip = P.recip
instance (AdditiveGroup a, MultiplicativeInvertible a) =>
MultiplicativeInvertible (Complex a) where
recip (rx :+ ix) = (rx `times` d) :+ (negate ix `times` d)
where
d = recip ((rx `times` rx) `plus` (ix `times` ix))
class ( MultiplicativeMagma b) =>
MultiplicativeHomomorphic a b where
timeshom :: a -> b
instance MultiplicativeMagma a => MultiplicativeHomomorphic a a where
timeshom a = a
class ( MultiplicativeUnital a
, MultiplicativeAssociative a) =>
MultiplicativeMonoidal a
instance MultiplicativeMonoidal Double
instance MultiplicativeMonoidal Float
instance MultiplicativeMonoidal Int
instance MultiplicativeMonoidal Integer
instance MultiplicativeMonoidal Bool
instance (AdditiveGroup a, MultiplicativeMonoidal a) =>
MultiplicativeMonoidal (Complex a)
class ( MultiplicativeCommutative a
, MultiplicativeUnital a
, MultiplicativeAssociative a) =>
Multiplicative a where
infixl 7 *
(*) :: a -> a -> a
a * b = times a b
instance Multiplicative Double
instance Multiplicative Float
instance Multiplicative Int
instance Multiplicative Integer
instance Multiplicative Bool
instance (AdditiveGroup a, Multiplicative a) =>
Multiplicative (Complex a) where
class ( MultiplicativeUnital a
, MultiplicativeAssociative a
, MultiplicativeInvertible a) =>
MultiplicativeLeftCancellative a where
infixl 7 ~/
(~/) :: a -> a -> a
a ~/ b = recip b `times` a
class ( MultiplicativeUnital a
, MultiplicativeAssociative a
, MultiplicativeInvertible a) =>
MultiplicativeRightCancellative a where
infixl 7 /~
(/~) :: a -> a -> a
a /~ b = a `times` recip b
class ( Multiplicative a
, MultiplicativeInvertible a) =>
MultiplicativeGroup a where
infixl 7 /
(/) :: a -> a -> a
(/) a b = a `times` recip b
instance MultiplicativeGroup Double
instance MultiplicativeGroup Float
instance (AdditiveGroup a, MultiplicativeGroup a) =>
MultiplicativeGroup (Complex a) where