Euclidean.hs

{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FlexibleInstances #-}

{- |
Copyright   :  (c) Henning Thielemann 2005-2010
License     :  GPL

Maintainer  :  numericprelude@henning-thielemann.de
Stability   :  provisional
Portability :  requires multi-parameter type classes

Abstraction of normed vector spaces
-}

module Algebra.NormedSpace.Euclidean where

import NumericPrelude.Base
import NumericPrelude.Numeric (sqr, abs, zero, (+), sum, Float, Double, Int, Integer, )

import qualified Number.Ratio as Ratio

import qualified Algebra.PrincipalIdealDomain as PID
import qualified Algebra.Algebraic as Algebraic
import qualified Algebra.Absolute      as Absolute
import qualified Algebra.Module    as Module

import qualified Data.Foldable as Fold


{-|
Helper class for 'C' that does not need an algebraic type @a@.

Minimal definition:
'normSqr'
-}
class (Absolute.C a, Module.C a v) => Sqr a v where
  {-| Square of the Euclidean norm of a vector.
      This is sometimes easier to implement. -}
  normSqr :: v -> a
--  normSqr = sqr . norm

{- |
Default definition for 'normSqr' that is based on 'Fold.Foldable' class.
-}
{-# INLINE normSqrFoldable #-}
normSqrFoldable ::
   (Sqr a v, Fold.Foldable f) => f v -> a
normSqrFoldable =
   Fold.foldl (\a v -> a + normSqr v) zero

{- |
Default definition for 'normSqr' that is based on 'Fold.Foldable' class
and the argument vector has at least one component.
-}
{-# INLINE normSqrFoldable1 #-}
normSqrFoldable1 ::
   (Sqr a v, Fold.Foldable f, Functor f) => f v -> a
normSqrFoldable1 =
   Fold.foldl1 (+) . fmap normSqr


{-|
A vector space equipped with an Euclidean or a Hilbert norm.

Minimal definition:
'norm'
-}
class (Sqr a v) => C a v where
  {-| Euclidean norm of a vector. -}
  norm :: v -> a


defltNorm :: (Algebraic.C a, Sqr a v) => v -> a
defltNorm = Algebraic.sqrt . normSqr


{-* Instances for atomic types -}

instance Sqr Float Float where
  normSqr = sqr

instance C Float Float where
  norm    = abs

instance Sqr Double Double where
  normSqr = sqr

instance C Double Double where
  norm    = abs

instance Sqr Int Int where
  normSqr = sqr

instance C Int Int where
  norm    = abs

instance Sqr Integer Integer where
  normSqr = sqr

instance C Integer Integer where
  norm    = abs


{-* Instances for composed types -}

instance (Absolute.C a, PID.C a) => Sqr (Ratio.T a) (Ratio.T a) where
  normSqr = sqr

instance (Sqr a v0, Sqr a v1) => Sqr a (v0, v1) where
  normSqr (x0,x1) = normSqr x0 + normSqr x1

instance (Algebraic.C a, Sqr a v0, Sqr a v1) => C a (v0, v1) where
  norm    = defltNorm

instance (Sqr a v0, Sqr a v1, Sqr a v2) => Sqr a (v0, v1, v2) where
  normSqr (x0,x1,x2) = normSqr x0 + normSqr x1 + normSqr x2

instance (Algebraic.C a, Sqr a v0, Sqr a v1, Sqr a v2) => C a (v0, v1, v2) where
  norm    = defltNorm

instance (Sqr a v) => Sqr a [v] where
  normSqr = sum . map normSqr

instance (Algebraic.C a, Sqr a v) => C a [v] where
  norm    = defltNorm