src/Normed.hs

   1 {-# LANGUAGE FlexibleInstances #-}
   2 {-# LANGUAGE RebindableSyntax #-}
   3
   4 -- | The 'Normed' class represents elements of a normed vector
   5 --   space. We define instances for all common numeric types.
   6 module Normed
   7 where
   8
   9 import BigFloat
  10
  11 import NumericPrelude hiding ( abs )
  12 import Algebra.Absolute ( abs )
  13 import qualified Algebra.Absolute as Absolute ( C )
  14 import qualified Algebra.Algebraic as Algebraic ( C )
  15 import Algebra.Algebraic ( root )
  16 import qualified Algebra.RealField as RealField ( C )
  17 import qualified Algebra.ToInteger as ToInteger ( C )
  18 import qualified Algebra.ToRational as ToRational ( C )
  19 import Data.Vector.Fixed ( S, Z )
  20 import qualified Data.Vector.Fixed as V (
  21   Arity,
  22   map,
  23   maximum )
  24 import Data.Vector.Fixed.Boxed ( Vec )
  25
  26 import Linear.Vector ( element_sum )
  27
  28
  29 -- | Instances of the 'Normed' class know how to compute their own
  30 --   p-norms for p=1,2,...,infinity.
  31 --
  32 class Normed a where
  33   norm_p :: (ToInteger.C c, Algebraic.C b, Absolute.C b) => c -> a -> b
  34   norm_infty :: (RealField.C b) => a -> b
  35
  36   -- | The \"usual\" norm. Defaults to the 2-norm.
  37   norm :: (Algebraic.C b, Absolute.C b) => a -> b
  38   norm = norm_p (2 :: Integer)
  39
  40 -- Define instances for common numeric types.
  41 instance Normed Integer where
  42   norm_p _ = abs . fromInteger
  43   norm_infty = abs . fromInteger
  44
  45 instance Normed Rational where
  46   norm_p _ = abs . fromRational'
  47   norm_infty = abs . fromRational'
  48
  49 instance Epsilon e => Normed (BigFloat e) where
  50   norm_p _ = abs . fromRational' . toRational
  51   norm_infty = abs . fromRational' . toRational
  52
  53 instance Normed Float where
  54   norm_p _ = abs . fromRational' . toRational
  55   norm_infty = abs . fromRational' . toRational
  56
  57 instance Normed Double where
  58   norm_p _ = abs . fromRational' . toRational
  59   norm_infty = abs . fromRational' . toRational
  60
  61
  62 -- | 'Normed' instance for vectors of length zero. These are easy.
  63 instance Normed (Vec Z a) where
  64   norm_p _ = const (fromInteger 0)
  65   norm_infty = const (fromInteger 0)
  66
  67
  68 -- | 'Normed' instance for vectors of length greater than zero. We
  69 --   need to know that the length is non-zero in order to invoke
  70 --   V.maximum. We will generally be working with n-by-1 /matrices/
  71 --   instead of vectors, but sometimes it's convenient to have these
  72 --   instances anyway.
  73 --
  74 --   Examples:
  75 --
  76 --   >>> import Data.Vector.Fixed (mk3)
  77 --   >>> import Linear.Vector (Vec3)
  78 --   >>> let b = mk3 1 2 3 :: Vec3 Double
  79 --   >>> norm_p 1 b :: Double
  80 --   6.0
  81 --   >>> norm b == sqrt 14
  82 --   True
  83 --   >>> norm_infty b :: Double
  84 --   3.0
  85 --
  86 instance (V.Arity n, Absolute.C a, ToRational.C a, Ord a)
  87          => Normed (Vec (S n) a) where
  88   norm_p p x =
  89     (root p') $ element_sum $ V.map element_function x
  90     where
  91       element_function y = fromRational' $ (toRational y)^p'
  92       p' = toInteger p
  93
  94   norm_infty x = fromRational' $ toRational $ V.maximum $ V.map abs x