X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FVector.hs;h=56bc2af84c1ffc71aa1e3646602e92b1d875e642;hb=c7b6b27f4304416dcec67519b710d090850c3caa;hp=7cc5e005ef22dab30af2ad8d0721550eecffd070;hpb=3c5015c938c96f70b15c6292198a01390ee6540a;p=numerical-analysis.git

diff --git a/src/Linear/Vector.hs b/src/Linear/Vector.hs
index 7cc5e00..56bc2af 100644
--- a/src/Linear/Vector.hs
+++ b/src/Linear/Vector.hs
@@ -1,212 +1,51 @@
 {-# LANGUAGE FlexibleContexts #-}
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE NoMonomorphismRestriction #-}
 {-# LANGUAGE ScopedTypeVariables #-}
 {-# LANGUAGE TypeFamilies #-}
 
-module Linear.Vector
+module Linear.Vector (
+  module Data.Vector.Fixed.Boxed,
+  Vec1,
+  (!?),
+  delete,
+  element_sum )
 where
 
-import Data.List (intercalate)
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Ring as Ring ( C )
 import Data.Vector.Fixed (
   Dim,
-  Fun(..),
-  N2,
-  N3,
-  N4,
+  N1,
+  S,
   Vector(..),
-  construct,
-  inspect,
-  toList,
-  )
-import qualified Data.Vector.Fixed as V (
-  eq,
+  fromList,
+  toList )
+import Data.Vector.Fixed (
+  (!),
   foldl,
-  length,
-  map,
-  replicate,
-  sum,
-  zipWith
-  )
-
-import Normed
-
--- | The Vn newtype simply wraps (Vector v a) so that we avoid
---   undecidable instances.
-newtype Vn v a = Vn (v a)
-
--- | Declare the dimension of the wrapper to be the dimension of what
---   it contains.
-type instance Dim (Vn v) = Dim v
-
-instance (Vector v a) => Vector (Vn v) a where
-  -- | Fortunately, 'Fun' is an instance of 'Functor'. The
-  --   'construct' defined on our contained type will return a
-  --   'Fun', and we simply slap our constructor on top with fmap.
-  construct = fmap Vn construct
-
-  -- | Defer to the inspect defined on the contained type.
-  inspect (Vn v1) = inspect v1
-
-instance (Show a, Vector v a) => Show (Vn v a) where
-  -- | Display vectors as ordinary tuples. This is poor practice, but
-  --   these results are primarily displayed interactively and
-  --   convenience trumps correctness (said the guy who insists his
-  --   vector lengths be statically checked at compile-time).
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (1,2)
-  --   >>> show v1
-  --   (1,2)
-  --
-  show (Vn v1) =
-    "(" ++ (intercalate "," element_strings) ++ ")"
-    where
-      v1l = toList v1
-      element_strings = Prelude.map show v1l
-
-
--- | We would really like to say, "anything that is a vector of
---   equatable things is itself equatable." The 'Vn' class
---   allows us to express this without a GHC battle.
---
---   Examples:
---
---   >>> let v1 = make2d (1,2)
---   >>> let v2 = make2d (1,2)
---   >>> let v3 = make2d (3,4)
---   >>> v1 == v2
---   True
---   >>> v1 == v3
---   False
---
-instance (Eq a, Vector v a) => Eq (Vn v a) where
-  (Vn v1) == (Vn v2) = v1 `V.eq` v2
-
+  length )
+import Data.Vector.Fixed.Boxed (
+  Vec,
+  Vec2,
+  Vec3,
+  Vec4,
+  Vec5 )
+import NumericPrelude hiding ( abs, length, foldl )
 
--- | The use of 'Num' here is of course incorrect (otherwise, we
---   wouldn't have to throw errors). But it's really nice to be able
---   to use normal addition/subtraction.
-instance (Num a, Vector v a) => Num (Vn v a) where
-  -- | Componentwise addition.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (1,2)
-  --   >>> let v2 = make2d (3,4)
-  --   >>> v1 + v2
-  --   (4,6)
-  --
-  (Vn v1) + (Vn v2) = Vn $ V.zipWith (+) v1 v2
 
-  -- | Componentwise subtraction.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (1,2)
-  --   >>> let v2 = make2d (3,4)
-  --   >>> v1 - v2
-  --   (-2,-2)
-  --
-  (Vn v1) - (Vn v2) = Vn $ V.zipWith (-) v1 v2
-
-  -- | Create an n-vector whose components are all equal to the given
-  --   integer. The result type must be specified since otherwise the
-  --   length n would be unknown.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = fromInteger 17 :: Vn Vec3 Int
-  --   (17,17,17)
-  --
-  fromInteger x = Vn $ V.replicate (fromInteger x)
-  (*) = error "multiplication of vectors is undefined"
-  abs = error "absolute value of vectors is undefined"
-  signum = error "signum of vectors is undefined"
-
-
--- | This is probably useless, since the vectors we usually contain
---   aren't functor instances.
-instance (Functor v) => Functor (Vn v) where
-  fmap f (Vn v1) = Vn (f `fmap` v1)
-
-
-instance (RealFloat a, Ord a, Vector v a) => Normed (Vn v a) where
-  -- | The infinity norm. We don't use V.maximum here because it
-  --   relies on a type constraint that the vector be non-empty and I
-  --   don't know how to pattern match it away.
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make3d (1,5,2)
-  --   >>> norm_infty v1
-  --   5
-  --
-  norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
-
-  -- | Generic p-norms. The usual norm in R^n is (norm_p 2).
-  --
-  --   Examples:
-  --
-  --   >>> let v1 = make2d (3,4)
-  --   >>> norm_p 1 v1
-  --   7.0
-  --   >>> norm_p 2 v1
-  --   5.0
-  --
-  norm_p p (Vn v1) =
-    realToFrac $ root $ V.sum $ V.map (exponentiate . abs) v1
-    where
-      exponentiate = (** (fromIntegral p))
-      root = (** (recip (fromIntegral p)))
-
--- | Dot (standard inner) product.
---
---   Examples:
---
---   >>> let v1 = make3d (1,2,3)
---   >>> let v2 = make3d (4,5,6)
---   >>> dot v1 v2
---   32
---
-dot :: (Num a, Vector v a) => Vn v a -> Vn v a -> a
-dot (Vn v1) (Vn v2) = V.sum $ V.zipWith (*) v1 v2
-
-
--- | The angle between @v1@ and @v2@ in Euclidean space.
---
---   Examples:
---
---   >>> let v1 = make2d (1.0, 0.0)
---   >>> let v2 = make2d (0.0, 1.0)
---   >>> angle v1 v2 == pi/2.0
---   True
---
-angle :: (RealFloat a, Vector v a) => Vn v a -> Vn v a -> a
-angle v1 v2 =
-  acos theta
-  where
-    theta = (v1 `dot` v2) / norms
-    norms = (norm v1) * (norm v2)
+type Vec1 = Vec N1
 
 
--- | Unsafe indexing.
---
---   Examples:
---
---   >>> let v1 = make2d (1,2)
---   >>> v1 ! 1
---   2
---
-(!) :: (Vector v a) => v a -> Int -> a
-(!) v1 idx = (toList v1) !! idx
 
 -- | Safe indexing.
 --
 --   Examples:
 --
---   >>> let v1 = make3d (1,2,3)
+--   >>> import Data.Vector.Fixed (mk3)
+--   >>> let v1 = mk3 1 2 3 :: Vec3 Int
 --   >>> v1 !? 2
 --   Just 3
 --   >>> v1 !? 3
@@ -214,92 +53,41 @@ angle v1 v2 =
 --
 (!?) :: (Vector v a) => v a -> Int -> Maybe a
 (!?) v1 idx
-  | idx < 0 || idx >= V.length v1 = Nothing
-  | otherwise                     = Just $ v1 ! idx
-
-
-
-
--- * Low-dimension vector wrappers.
---
--- These wrappers are instances of 'Vector', so they inherit all of
--- the userful instances defined above. But, they use fixed
--- constructors, so you can pattern match out the individual
--- components.
-
--- | Convenient constructor for 2D vectors.
---
---   Examples:
---
---   >>> import Roots.Simple
---   >>> let h = 0.5 :: Double
---   >>> let g1 (Vn (Vec2D x y)) = 1.0 + h*exp(-(x^2))/(1.0 + y^2)
---   >>> let g2 (Vn (Vec2D x y)) = 0.5 + h*atan(x^2 + y^2)
---   >>> let g u = make2d ((g1 u), (g2 u))
---   >>> let u0 = make2d (1.0, 1.0)
---   >>> let eps = 1/(10^9)
---   >>> fixed_point g eps u0
---   (1.0728549599342185,1.0820591495686167)
---
-data Vec2D a = Vec2D a a
-type instance Dim Vec2D = N2
-instance Vector Vec2D a where
-  inspect (Vec2D x y) (Fun f) = f x y
-  construct = Fun Vec2D
-
-data Vec3D a = Vec3D a a a
-type instance Dim Vec3D = N3
-instance Vector Vec3D a where
-  inspect (Vec3D x y z) (Fun f) = f x y z
-  construct = Fun Vec3D
-
-data Vec4D a = Vec4D a a a a
-type instance Dim Vec4D = N4
-instance Vector Vec4D a where
-  inspect (Vec4D w x y z) (Fun f) = f w x y z
-  construct = Fun Vec4D
-
+  | idx < 0 || idx >= length v1 = Nothing
+  | otherwise                   = Just $ v1 ! idx
 
--- | Convenience function for creating 2d vectors.
---
---   Examples:
---
---   >>> let v1 = make2d (1,2)
---   >>> v1
---   (1,2)
---   >>> let Vn (Vec2D x y) = v1
---   >>> (x,y)
---   (1,2)
---
-make2d :: forall a. (a,a) -> Vn Vec2D a
-make2d (x,y) = Vn (Vec2D x y)
 
-
--- | Convenience function for creating 3d vectors.
+-- | Remove an element of the given vector.
 --
 --   Examples:
 --
---   >>> let v1 = make3d (1,2,3)
---   >>> v1
---   (1,2,3)
---   >>> let Vn (Vec3D x y z) = v1
---   >>> (x,y,z)
---   (1,2,3)
---
-make3d :: forall a. (a,a,a) -> Vn Vec3D a
-make3d (x,y,z) = Vn (Vec3D x y z)
+--   >>> import Data.Vector.Fixed (mk3)
+--   >>> let b = mk3 1 2 3 :: Vec3 Int
+--   >>> delete b 1 :: Vec2 Int
+--   fromList [1,3]
+--
+delete :: (Vector v a,
+           Vector w a,
+           Dim v ~ S (Dim w))
+       => v a
+       -> Int
+       -> w a
+delete v1 idx =
+  fromList (lhalf ++ rhalf')
+  where
+    (lhalf, rhalf) = splitAt idx (toList v1)
+    rhalf' = tail rhalf
 
 
--- | Convenience function for creating 4d vectors.
+-- | We provide our own sum because sum relies on a Num instance
+--   from the Prelude that we don't have.
 --
 --   Examples:
 --
---   >>> let v1 = make4d (1,2,3,4)
---   >>> v1
---   (1,2,3,4)
---   >>> let Vn (Vec4D w x y z) = v1
---   >>> (w,x,y,z)
---   (1,2,3,4)
+--   >>> import Data.Vector.Fixed (mk3)
+--   >>> let b = mk3 1 2 3 :: Vec3 Int
+--   >>> element_sum b
+--   6
 --
-make4d :: forall a. (a,a,a,a) -> Vn Vec4D a
-make4d (w,x,y,z) = Vn (Vec4D w x y z)
+element_sum :: (Additive.C a, Ring.C a, Vector v a) => v a -> a
+element_sum = foldl (+) (fromInteger 0)