-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE RebindableSyntax #-}
module Linear.Matrix
where
+import Data.List (intercalate)
+
import Data.Vector.Fixed (
Dim,
Vector
)
import qualified Data.Vector.Fixed as V (
+ Fun(..),
+ N1,
+ and,
+ eq,
+ foldl,
fromList,
length,
map,
- toList
+ maximum,
+ replicate,
+ toList,
+ zipWith
)
-import Data.Vector.Fixed.Internal (arity)
-
+import Data.Vector.Fixed.Internal (Arity, arity, S, Dim)
import Linear.Vector
+import Normed
+
+import NumericPrelude hiding (abs)
+import qualified Algebra.Algebraic as Algebraic
+import qualified Algebra.Absolute as Absolute
+import qualified Algebra.Additive as Additive
+import qualified Algebra.Ring as Ring
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.RealRing as RealRing
+import qualified Algebra.ToRational as ToRational
+import qualified Algebra.Transcendental as Transcendental
+import qualified Prelude as P
+
+data Mat v w a = (Vector v (w a), Vector w a) => Mat (v (w a))
+type Mat1 a = Mat D1 D1 a
+type Mat2 a = Mat D2 D2 a
+type Mat3 a = Mat D3 D3 a
+type Mat4 a = Mat D4 D4 a
+
+-- We can't just declare that all instances of Vector are instances of
+-- Eq unfortunately. We wind up with an overlapping instance for
+-- w (w a).
+instance (Eq a, Vector v Bool, Vector w Bool) => Eq (Mat v w a) where
+ -- | Compare a row at a time.
+ --
+ -- Examples:
+ --
+ -- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> let m2 = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> let m3 = fromList [[5,6],[7,8]] :: Mat2 Int
+ -- >>> m1 == m2
+ -- True
+ -- >>> m1 == m3
+ -- False
+ --
+ (Mat rows1) == (Mat rows2) =
+ V.and $ V.zipWith comp rows1 rows2
+ where
+ -- Compare a row, one column at a time.
+ comp row1 row2 = V.and (V.zipWith (==) row1 row2)
+
+
+instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where
+ -- | Display matrices and 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 m = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> show m
+ -- ((1,2),(3,4))
+ --
+ show (Mat rows) =
+ "(" ++ (intercalate "," (V.toList row_strings)) ++ ")"
+ where
+ row_strings = V.map show_vector rows
+ show_vector v1 =
+ "(" ++ (intercalate "," element_strings) ++ ")"
+ where
+ v1l = V.toList v1
+ element_strings = P.map show v1l
+
-type Mat v w a = Vn v (Vn w a)
-type Mat2 a = Mat Vec2D Vec2D a
-type Mat3 a = Mat Vec3D Vec3D a
-type Mat4 a = Mat Vec4D Vec4D a
-- | Convert a matrix to a nested list.
-toList :: (Vector v (Vn w a), Vector w a) => Mat v w a -> [[a]]
-toList m = map V.toList (V.toList m)
+toList :: Mat v w a -> [[a]]
+toList (Mat rows) = map V.toList (V.toList rows)
-- | Create a matrix from a nested list.
-fromList :: (Vector v (Vn w a), Vector w a) => [[a]] -> Mat v w a
-fromList vs = V.fromList $ map V.fromList vs
+fromList :: (Vector v (w a), Vector w a, Vector v a) => [[a]] -> Mat v w a
+fromList vs = Mat (V.fromList $ map V.fromList vs)
-- | Unsafe indexing.
-(!!!) :: (Vector v (Vn w a), Vector w a) => Mat v w a -> (Int, Int) -> a
+(!!!) :: (Vector w a) => Mat v w a -> (Int, Int) -> a
(!!!) m (i, j) = (row m i) ! j
-- | Safe indexing.
-(!!?) :: (Vector v (Vn w a), Vector w a) => Mat v w a
- -> (Int, Int)
- -> Maybe a
-(!!?) m (i, j)
+(!!?) :: Mat v w a -> (Int, Int) -> Maybe a
+(!!?) m@(Mat rows) (i, j)
| i < 0 || j < 0 = Nothing
- | i > V.length m = Nothing
+ | i > V.length rows = Nothing
| otherwise = if j > V.length (row m j)
then Nothing
else Just $ (row m j) ! j
-- | The number of rows in the matrix.
-nrows :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-nrows = V.length
+nrows :: Mat v w a -> Int
+nrows (Mat rows) = V.length rows
-- | The number of columns in the first row of the
-- matrix. Implementation stolen from Data.Vector.Fixed.length.
-ncols :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-ncols _ = arity (undefined :: Dim w)
+ncols :: forall v w a. (Vector w a) => Mat v w a -> Int
+ncols _ = (arity (undefined :: Dim w))
-- | Return the @i@th row of @m@. Unsafe.
-row :: (Vector v (Vn w a), Vector w a) => Mat v w a
- -> Int
- -> Vn w a
-row m i = m ! i
+row :: Mat v w a -> Int -> w a
+row (Mat rows) i = rows ! i
-- | Return the @j@th column of @m@. Unsafe.
-column :: (Vector v a, Vector v (Vn w a), Vector w a) => Mat v w a
- -> Int
- -> Vn v a
-column m j =
- V.map (element j) m
+column :: (Vector v a) => Mat v w a -> Int -> v a
+column (Mat rows) j =
+ V.map (element j) rows
where
element = flip (!)
-- >>> transpose m
-- ((1,3),(2,4))
--
-transpose :: (Vector v (Vn w a),
- Vector w (Vn v a),
+transpose :: (Vector w (v a),
Vector v a,
Vector w a)
=> Mat v w a
-> Mat w v a
-transpose m = V.fromList column_list
+transpose m = Mat $ V.fromList column_list
where
column_list = [ column m i | i <- [0..(ncols m)-1] ]
+
-- | Is @m@ symmetric?
--
-- Examples:
-- >>> symmetric m2
-- False
--
-symmetric :: (Vector v (Vn w a),
+symmetric :: (Vector v (w a),
Vector w a,
v ~ w,
Vector w Bool,
-- ((0,1,2),(1,2,3),(2,3,4))
--
construct :: forall v w a.
- (Vector v (Vn w a),
+ (Vector v (w a),
Vector w a)
=> (Int -> Int -> a)
-> Mat v w a
-construct lambda = rows
+construct lambda = Mat rows
where
-- The arity trick is used in Data.Vector.Fixed.length.
imax = (arity (undefined :: Dim v)) - 1
-- ((20.000000000000004,-1.0),(-1.0,20.0))
--
cholesky :: forall a v w.
- (RealFloat a,
- Vector v (Vn w a),
- Vector w a)
+ (Algebraic.C a,
+ Vector v (w a),
+ Vector w a,
+ Vector v a)
=> (Mat v w a)
-> (Mat v w a)
cholesky m = construct r
where
r :: Int -> Int -> a
- r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)**2 | k <- [0..i-1]])
+ r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)^2 | k <- [0..i-1]])
| i < j =
(((m !!! (i,j)) - sum [(r k i)*(r k j) | k <- [0..i-1]]))/(r i i)
| otherwise = 0
+
-- | Matrix multiplication. Our 'Num' instance doesn't define one, and
-- we need additional restrictions on the result type anyway.
--
-- >>> m1 `mult` m2
-- ((22,28),(49,64))
--
-mult :: (Num a,
- Vector v (Vn w a),
+mult :: (Ring.C a,
+ Vector v a,
Vector w a,
- Vector w (Vn z a),
Vector z a,
- Vector v (Vn z a))
+ Vector v (z a))
=> Mat v w a
-> Mat w z a
-> Mat v z a
where
lambda i j =
sum [(m1 !!! (i,k)) * (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
+
+
+
+instance (Ring.C a,
+ Vector v (w a),
+ Vector w a)
+ => Additive.C (Mat v w a) where
+
+ (Mat rows1) + (Mat rows2) =
+ Mat $ V.zipWith (V.zipWith (+)) rows1 rows2
+
+ (Mat rows1) - (Mat rows2) =
+ Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
+
+ zero = Mat (V.replicate $ V.replicate (fromInteger 0))
+
+
+instance (Ring.C a,
+ Vector v (w a),
+ Vector w a,
+ v ~ w)
+ => Ring.C (Mat v w a) where
+ one = Mat (V.replicate $ V.replicate (fromInteger 1))
+ m1 * m2 = m1 `mult` m2
+
+
+instance (Algebraic.C a,
+ ToRational.C a,
+ Vector v (w a),
+ Vector w a,
+ Vector v a,
+ Vector v [a])
+ => Normed (Mat v w a) where
+ -- Treat the matrix as a big vector.
+ norm_p p (Mat rows) =
+ sqrt $ sum [(fromRational' $ toRational x)^2 | x <- xs]
+ where
+ xs = concat $ V.toList $ V.map V.toList rows
+
+ norm_infty m@(Mat rows)
+ | nrows m == 0 || ncols m == 0 = 0
+ | otherwise =
+ fromRational' $ toRational $
+ P.maximum $ V.toList $ V.map (P.maximum . V.toList) rows
+
+
+
+
+
+-- Vector helpers. We want it to be easy to create low-dimension
+-- column vectors.
+type Vec a b = Mat a D1 b
+
+vec2d :: (a,a) -> Mat D2 D1 a
+vec2d (x,y) = Mat (D2 (D1 x) (D1 y))
+
+vec3d :: (a,a,a) -> Mat D3 D1 a
+vec3d (x,y,z) = Mat (D3 (D1 x) (D1 y) (D1 z))
+
+vec4d :: (a,a,a,a) -> Mat D4 D1 a
+vec4d (w,x,y,z) = Mat (D4 (D1 w) (D1 x) (D1 y) (D1 z))
+
+dot :: (RealRing.C a,
+ Dim w ~ V.N1,
+ Vector v a,
+ Vector w a,
+ Vector w (v a),
+ Vector w (w a))
+ => Mat v w a
+ -> Mat v w a
+ -> a
+v1 `dot` v2 = ((transpose v1) `mult` v2) !!! (0, 0)
+
+
+-- | 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 :: (Transcendental.C a,
+ RealRing.C a,
+ Dim w ~ V.N1,
+ Vector w (w a),
+ Vector v [a],
+ Vector v a,
+ Vector w a,
+ Vector v (w a),
+ Vector w (v a),
+ ToRational.C a)
+ => Mat v w a
+ -> Mat v w a
+ -> a
+angle v1 v2 =
+ acos theta
+ where
+ theta = (recip norms) * (v1 `dot` v2)
+ norms = (norm v1) * (norm v2)
import Data.Vector.Fixed (
Dim,
Fun(..),
+ N1,
N2,
N3,
N4,
toList,
)
import qualified Data.Vector.Fixed as V (
- eq,
- 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
+-- * 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.
-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
+data D1 a = D1 a
+type instance Dim D1 = N1
+instance Vector D1 a where
+ inspect (D1 x) (Fun f) = f x
+ construct = Fun D1
- -- | Defer to the inspect defined on the contained type.
- inspect (Vn v1) = inspect v1
+data D2 a = D2 a a
+type instance Dim D2 = N2
+instance Vector D2 a where
+ inspect (D2 x y) (Fun f) = f x y
+ construct = Fun D2
-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
+data D3 a = D3 a a a
+type instance Dim D3 = N3
+instance Vector D3 a where
+ inspect (D3 x y z) (Fun f) = f x y z
+ construct = Fun D3
+
+data D4 a = D4 a a a a
+type instance Dim D4 = N4
+instance Vector D4 a where
+ inspect (D4 w x y z) (Fun f) = f w x y z
+ construct = Fun D4
--- | 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.
+-- | Unsafe indexing.
--
-- Examples:
--
--- >>> let v1 = make2d (1,2)
--- >>> let v2 = make2d (1,2)
--- >>> let v3 = make2d (3,4)
--- >>> v1 == v2
--- True
--- >>> v1 == v3
--- False
+-- >>> let v1 = Vec2D 1 2
+-- >>> v1 ! 1
+-- 2
--
-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
-
- -- | 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
+(!) :: (Vector v a) => v a -> Int -> a
+(!) v1 idx = (toList v1) !! idx
- -- | 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"
+-- | Safe indexing.
+--
+-- Examples:
+--
+-- >>> let v1 = Vec3D 1 2 3
+-- >>> v1 !? 2
+-- Just 3
+-- >>> v1 !? 3
+-- Nothing
+--
+(!?) :: (Vector v a) => v a -> Int -> Maybe a
+(!?) v1 idx
+ | idx < 0 || idx >= V.length v1 = Nothing
+ | otherwise = Just $ v1 ! idx
--- | 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
+--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.
-- >>> norm_infty v1
-- 5
--
- norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
+-- norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
-- | Generic p-norms. The usual norm in R^n is (norm_p 2).
--
-- >>> 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)
+-- norm_p p (Vn v1) =
+-- realToFrac $ root $ V.sum $ V.map (exponentiate . abs) v1
+-- where
+-- exponentiate = (** (fromIntegral p))
+-- root = (** (recip (fromIntegral p)))
--- | 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)
--- >>> v1 !? 2
--- Just 3
--- >>> v1 !? 3
--- Nothing
---
-(!?) :: (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:
-- >>> 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
-
-
--- | 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.
---
--- 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)
-
-
--- | Convenience function for creating 4d vectors.
---
--- 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)
---
-make4d :: forall a. (a,a,a,a) -> Vn Vec4D a
-make4d (w,x,y,z) = Vn (Vec4D w x y z)
projects / numerical-analysis.git / commitdiff