{-# 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,
+ Vec0,
+ 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,
+ Z,
+ 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
-
-
--- | 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
+ length )
+import Data.Vector.Fixed.Boxed (
+ Vec,
+ Vec2,
+ Vec3,
+ Vec4,
+ Vec5 )
+import NumericPrelude hiding ( abs, length, foldl )
- -- | 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
+type Vec0 = Vec Z
+type Vec1 = Vec N1
- -- | 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)
-
-
--- | 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
--
(!?) :: (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)
projects / numerical-analysis.git / blobdiff