src/FixedVector.hs

   1 {-# LANGUAGE FlexibleContexts #-}
   2 {-# LANGUAGE FlexibleInstances #-}
   3 {-# LANGUAGE MultiParamTypeClasses #-}
   4 {-# LANGUAGE ScopedTypeVariables #-}
   5 {-# LANGUAGE TypeFamilies #-}
   6
   7 module FixedVector
   8 where
   9
  10 import Data.List (intercalate)
  11 import qualified Data.Vector.Fixed as V
  12
  13 import Normed
  14
  15 -- | The Vn newtype simply wraps (Vector v a) so that we avoid
  16 --   undecidable instances.
  17 newtype Vn a = Vn a
  18
  19
  20 instance (Show a, V.Vector v a) => Show (Vn (v a)) where
  21   -- | Display vectors as ordinary tuples. This is poor practice, but
  22   --   these results are primarily displayed interactively and
  23   --   convenience trumps correctness (said the guy who insists his
  24   --   vector lengths be statically checked at compile-time).
  25   --
  26   --   Examples:
  27   --
  28   --   >>> let v1 = make2d (1,2)
  29   --   >>> show v1
  30   --   (1,2)
  31   --
  32   show (Vn v1) =
  33     "(" ++ (intercalate "," element_strings) ++ ")"
  34     where
  35       v1l = V.toList v1
  36       element_strings = Prelude.map show v1l
  37
  38
  39 -- | We would really like to say, "anything that is a vector of
  40 --   equatable things is itself equatable." The 'Vn' class
  41 --   allows us to express this without a GHC battle.
  42 --
  43 --   Examples:
  44 --
  45 --   >>> let v1 = make2d (1,2)
  46 --   >>> let v2 = make2d (1,2)
  47 --   >>> let v3 = make2d (3,4)
  48 --   >>> v1 == v2
  49 --   True
  50 --   >>> v1 == v3
  51 --   False
  52 --
  53 instance (Eq a, V.Vector v a, V.Vector v Bool) => Eq (Vn (v a)) where
  54   (Vn v1) == (Vn v2) = V.foldl (&&) True (V.zipWith (==) v1 v2)
  55
  56
  57 -- | The use of 'Num' here is of course incorrect (otherwise, we
  58 --   wouldn't have to throw errors). But it's really nice to be able
  59 --   to use normal addition/subtraction.
  60 instance (Num a, V.Vector v a) => Num (Vn (v a)) where
  61   -- | Componentwise addition.
  62   --
  63   --   Examples:
  64   --
  65   --   >>> let v1 = make2d (1,2)
  66   --   >>> let v2 = make2d (3,4)
  67   --   >>> v1 + v2
  68   --   (4,6)
  69   --
  70   (Vn v1) + (Vn v2) = Vn $ V.zipWith (+) v1 v2
  71
  72   -- | Componentwise subtraction.
  73   --
  74   --   Examples:
  75   --
  76   --   >>> let v1 = make2d (1,2)
  77   --   >>> let v2 = make2d (3,4)
  78   --   >>> v1 - v2
  79   --   (-2,-2)
  80   --
  81   (Vn v1) - (Vn v2) = Vn $ V.zipWith (-) v1 v2
  82
  83   -- | Create an n-vector whose components are all equal to the given
  84   --   integer. The result type must be specified since otherwise the
  85   --   length n would be unknown.
  86   --
  87   --   Examples:
  88   --
  89   --   >>> let v1 = fromInteger 17 :: Vn (Vec3 Int)
  90   --   (17,17,17)
  91   --
  92   fromInteger x = Vn $ V.replicate (fromInteger x)
  93   (*) = error "multiplication of vectors is undefined"
  94   abs = error "absolute value of vectors is undefined"
  95   signum = error "signum of vectors is undefined"
  96
  97 instance Functor Vn where
  98   fmap f (Vn v1) = Vn (f v1)
  99
 100 instance (RealFloat a, Ord a, V.Vector v a) => Normed (Vn (v a)) where
 101   -- | The infinity norm. We don't use V.maximum here because it
 102   --   relies on a type constraint that the vector be non-empty and I
 103   --   don't know how to pattern match it away.
 104   --
 105   --   Examples:
 106   --
 107   --   >>> let v1 = make3d (1,5,2)
 108   --   >>> norm_infty v1
 109   --   5
 110   --
 111   norm_infty (Vn v1) = fromRational $ toRational $ V.foldl max 0 v1
 112
 113   -- | Generic p-norms. The usual norm in R^n is (norm_p 2).
 114   --
 115   --   Examples:
 116   --
 117   --   >>> let v1 = make2d (3,4)
 118   --   >>> norm_p 1 v1
 119   --   7.0
 120   --   >>> norm_p 2 v1
 121   --   5.0
 122   --
 123   norm_p p (Vn v1) =
 124     fromRational $ toRational $ root $ V.sum $ V.map (exponentiate . abs) v1
 125     where
 126       exponentiate = (** (fromIntegral p))
 127       root = (** (recip (fromIntegral p)))
 128
 129 -- | Dot (standard inner) product.
 130 --
 131 --   Examples:
 132 --
 133 --   >>> let v1 = make3d (1,2,3)
 134 --   >>> let v2 = make3d (4,5,6)
 135 --   >>> dot v1 v2
 136 --   32
 137 --
 138 dot :: (Num a, V.Vector v a) => Vn (v a) -> Vn (v a) -> a
 139 dot (Vn v1) (Vn v2) = V.sum $ V.zipWith (*) v1 v2
 140
 141
 142 -- | The angle between @v1@ and @v2@ in Euclidean space.
 143 --
 144 --   Examples:
 145 --
 146 --   >>> let v1 = make2d (1.0, 0.0)
 147 --   >>> let v2 = make2d (0.0, 1.0)
 148 --   >>> angle v1 v2 == pi/2.0
 149 --   True
 150 --
 151 angle :: (RealFloat a, V.Vector v a) => Vn (v a) -> Vn (v a) -> a
 152 angle v1 v2 =
 153   acos theta
 154   where
 155     theta = (v1 `dot` v2) / norms
 156     norms = (norm v1) * (norm v2)
 157
 158 -- | Unsafe indexing.
 159 --
 160 --   Examples:
 161 --
 162 --   >>> let v1 = make3d (1,2,3)
 163 --   >>> v1 ! 2
 164 --   3
 165 --   >>> v1 ! 3
 166 --   *** Exception: Data.Vector.Fixed.!: index out of range
 167 --
 168 (!) :: (V.Vector v a) => Vn (v a) -> Int -> a
 169 (!) (Vn v1) idx = v1 V.! idx
 170
 171
 172 -- | Safe indexing.
 173 --   Examples:
 174 --
 175 --   >>> let v1 = make3d (1,2,3)
 176 --   >>> v1 !? 2
 177 --   Just 3
 178 --   >>> v1 !? 3
 179 --   Nothing
 180 --
 181 (!?) :: (V.Vector v a) => Vn (v a) -> Int -> Maybe a
 182 (!?) v1@(Vn v2) idx
 183   | idx < 0 || idx >= V.length v2 = Nothing
 184   | otherwise                     = Just $ v1 ! idx
 185
 186
 187 -- * Two- and three-dimensional wrappers.
 188 --
 189 -- These two wrappers are instances of 'Vector', so they inherit all
 190 -- of the userful instances defined above. But, they use fixed
 191 -- constructors, so you can pattern match out the individual
 192 -- components.
 193
 194 data Vec2D a = Vec2D a a
 195 type instance V.Dim Vec2D = V.N2
 196 instance V.Vector Vec2D a where
 197   inspect (Vec2D x y) (V.Fun f) = f x y
 198   construct = V.Fun Vec2D
 199
 200 data Vec3D a = Vec3D a a a
 201 type instance V.Dim Vec3D = V.N3
 202 instance V.Vector Vec3D a where
 203   inspect (Vec3D x y z) (V.Fun f) = f x y z
 204   construct = V.Fun Vec3D
 205
 206
 207 -- | Convenience function for creating 2d vectors.
 208 --
 209 --   Examples:
 210 --
 211 --   >>> let v1 = make2d (1,2)
 212 --   >>> v1
 213 --   (1,2)
 214 --   >>> let Vn (Vec2D x y) = v1
 215 --   >>> (x,y)
 216 --   (1,2)
 217 --
 218 make2d :: forall a. (a,a) -> Vn (Vec2D a)
 219 make2d (x,y) = Vn (Vec2D x y)
 220
 221
 222 -- | Convenience function for creating 3d vectors.
 223 --
 224 --   Examples:
 225 --
 226 --   >>> let v1 = make3d (1,2,3)
 227 --   >>> v1
 228 --   (1,2,3)
 229 --   >>> let Vn (Vec3D x y z) = v1
 230 --   >>> (x,y,z)
 231 --   (1,2,3)
 232 --
 233 make3d :: forall a. (a,a,a) -> Vn (Vec3D a)
 234 make3d (x,y,z) = Vn (Vec3D x y z)