||| Instances of algebraic classes (group, ring, etc) for numeric data types, ||| and Complex number types. module Control.Algebra.NumericInstances import Control.Algebra import Control.Algebra.VectorSpace import Data.Complex import Data.ZZ instance Semigroup Integer where (<+>) = (+) instance Monoid Integer where neutral = 0 instance Group Integer where inverse = (* -1) instance AbelianGroup Integer instance Ring Integer where (<.>) = (*) instance RingWithUnity Integer where unity = 1 instance Semigroup Int where (<+>) = (+) instance Monoid Int where neutral = 0 instance Group Int where inverse = (* -1) instance AbelianGroup Int instance Ring Int where (<.>) = (*) instance RingWithUnity Int where unity = 1 instance Semigroup Float where (<+>) = (+) instance Monoid Float where neutral = 0 instance Group Float where inverse = (* -1) instance AbelianGroup Float instance Ring Float where (<.>) = (*) instance RingWithUnity Float where unity = 1 instance Field Float where inverseM f _ = 1 / f instance Semigroup Nat where (<+>) = (+) instance Monoid Nat where neutral = 0 instance Semigroup ZZ where (<+>) = (+) instance Monoid ZZ where neutral = 0 instance Group ZZ where inverse = (* -1) instance AbelianGroup ZZ instance Ring ZZ where (<.>) = (*) instance RingWithUnity ZZ where unity = 1 instance Semigroup a => Semigroup (Complex a) where (<+>) (a :+ b) (c :+ d) = (a <+> c) :+ (b <+> d) instance Monoid a => Monoid (Complex a) where neutral = (neutral :+ neutral) instance Group a => Group (Complex a) where inverse (r :+ i) = (inverse r :+ inverse i) instance Ring a => AbelianGroup (Complex a) where {} instance Ring a => Ring (Complex a) where (<.>) (a :+ b) (c :+ d) = (a <.> c <-> b <.> d) :+ (a <.> d <+> b <.> c) instance RingWithUnity a => RingWithUnity (Complex a) where unity = (unity :+ neutral) instance RingWithUnity a => Module a (Complex a) where (<#>) x = map (x <.>) instance RingWithUnity a => InnerProductSpace a (Complex a) where (x :+ y) <||> z = realPart \$ (x :+ inverse y) <.> z