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 -- Ensure that we don't use the Lattice.C "max" function, that
  12 -- only works on Integer/Bool.
  13 import NumericPrelude hiding ( abs, max )
  14 import Algebra.Absolute ( abs )
  15 import qualified Algebra.Absolute as Absolute ( C )
  16 import qualified Algebra.Algebraic as Algebraic ( C )
  17 import Algebra.Algebraic ( root )
  18 import qualified Algebra.RealField as RealField ( C )
  19 import qualified Algebra.ToInteger as ToInteger ( C )
  20 import qualified Algebra.ToRational as ToRational ( C )
  21 import qualified Data.Vector.Fixed as V (
  22   Arity,
  23   foldl,
  24   map )
  25 import Data.Vector.Fixed.Boxed ( Vec )
  26 import qualified Prelude as P ( max )
  27 import Linear.Vector ( element_sum )
  28
  29
  30 -- | Instances of the 'Normed' class know how to compute their own
  31 --   p-norms for p=1,2,...,infinity.
  32 --
  33 class Normed a where
  34   norm_p :: (ToInteger.C c, Algebraic.C b, Absolute.C b) => c -> a -> b
  35   norm_infty :: (RealField.C b) => a -> b
  36
  37   -- | The \"usual\" norm. Defaults to the 2-norm.
  38   norm :: (Algebraic.C b, Absolute.C b) => a -> b
  39   norm = norm_p (2 :: Integer)
  40
  41 -- Define instances for common numeric types.
  42 instance Normed Integer where
  43   norm_p _ = abs . fromInteger
  44   norm_infty = abs . fromInteger
  45
  46 instance Normed Rational where
  47   norm_p _ = abs . fromRational'
  48   norm_infty = abs . fromRational'
  49
  50 instance Epsilon e => Normed (BigFloat e) where
  51   norm_p _ = abs . fromRational' . toRational
  52   norm_infty = abs . fromRational' . toRational
  53
  54 instance Normed Float where
  55   norm_p _ = abs . fromRational' . toRational
  56   norm_infty = abs . fromRational' . toRational
  57
  58 instance Normed Double where
  59   norm_p _ = abs . fromRational' . toRational
  60   norm_infty = abs . fromRational' . toRational
  61
  62
  63 -- | 'Normed' instance for vectors of any length. We will generally be
  64 --   working with n-by-1 /matrices/ instead of vectors, but sometimes
  65 --   it's convenient to have these instances anyway.
  66 --
  67 --   Examples:
  68 --
  69 --   >>> import Data.Vector.Fixed (mk3)
  70 --   >>> import Linear.Vector (Vec0, Vec3)
  71 --   >>> let b = mk3 1 2 3 :: Vec3 Double
  72 --   >>> norm_p 1 b :: Double
  73 --   6.0
  74 --   >>> norm b == sqrt 14
  75 --   True
  76 --   >>> norm_infty b :: Double
  77 --   3.0
  78 --
  79 --   >>> let b = undefined :: Vec0 Int
  80 --   >>> norm b
  81 --   0.0
  82 --
  83 instance (V.Arity n, Absolute.C a, ToRational.C a, Ord a)
  84          => Normed (Vec n a) where
  85   norm_p p x =
  86     (root p') $ element_sum $ V.map element_function x
  87     where
  88       element_function y = fromRational' $ (toRational y)^p'
  89       p' = toInteger p
  90
  91   norm_infty x = fromRational' $ toRational $ (V.foldl P.max 0) $ V.map abs x