-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE RebindableSyntax #-}
+-- | Boxed matrices; that is, boxed m-vectors of boxed n-vectors. We
+-- assume that the underlying representation is
+-- Data.Vector.Fixed.Boxed.Vec for simplicity. It was tried in
+-- generality and failed.
+--
module Linear.Matrix
where
+import Data.List (intercalate)
+
import Data.Vector.Fixed (
- Dim,
- Vector
+ (!),
+ N1,
+ N2,
+ N3,
+ N4,
+ N5,
+ S,
+ Z,
+ generate,
+ mk1,
+ mk2,
+ mk3,
+ mk4,
+ mk5
)
import qualified Data.Vector.Fixed as V (
+ and,
fromList,
+ head,
length,
map,
- toList
+ maximum,
+ replicate,
+ toList,
+ zipWith
)
-import Data.Vector.Fixed.Internal (arity)
-
+import Data.Vector.Fixed.Boxed (Vec)
+import Data.Vector.Fixed.Cont (Arity, arity)
import Linear.Vector
+import Normed
+
+import NumericPrelude hiding ((*), abs)
+import qualified NumericPrelude as NP ((*))
+import qualified Algebra.Algebraic as Algebraic
+import Algebra.Algebraic (root)
+import qualified Algebra.Additive as Additive
+import qualified Algebra.Ring as Ring
+import qualified Algebra.Module as Module
+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 m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
+type Mat1 a = Mat N1 N1 a
+type Mat2 a = Mat N2 N2 a
+type Mat3 a = Mat N3 N3 a
+type Mat4 a = Mat N4 N4 a
+type Mat5 a = Mat N5 N5 a
+
+instance (Eq a) => Eq (Mat m n 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) => Show (Mat m n 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 m n 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 :: (Arity m, Arity n) => [[a]] -> Mat m n 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
+(!!!) :: (Arity m, Arity n) => Mat m n 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 m n 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 :: forall m n a. (Arity m) => Mat m n a -> Int
+nrows _ = arity (undefined :: m)
-- | 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 m n a. (Arity n) => Mat m n a -> Int
+ncols _ = arity (undefined :: n)
+
-- | 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 m n a -> Int -> (Vec n 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 :: Mat m n a -> Int -> (Vec m a)
+column (Mat rows) j =
+ V.map (element j) rows
where
element = flip (!)
+
+
-- | Transpose @m@; switch it's columns and its rows. This is a dirty
-- implementation.. it would be a little cleaner to use imap, but it
-- doesn't seem to work.
-- >>> transpose m
-- ((1,3),(2,4))
--
-transpose :: (Vector v (Vn w a),
- Vector w (Vn v a),
- Vector v a,
- Vector w a)
- => Mat v w a
- -> Mat w v a
-transpose m = V.fromList column_list
+transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
+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),
- Vector w a,
- v ~ w,
- Vector w Bool,
- Eq a)
- => Mat v w a
- -> Bool
+symmetric :: (Eq a, Arity m) => Mat m m a -> Bool
symmetric m =
m == (transpose m)
-- entries in the matrix. The i,j entry of the resulting matrix will
-- have the value returned by lambda i j.
--
--- TODO: Don't cheat with fromList.
---
-- Examples:
--
-- >>> let lambda i j = i + j
-- >>> construct lambda :: Mat3 Int
-- ((0,1,2),(1,2,3),(2,3,4))
--
-construct :: forall v w a.
- (Vector v (Vn w a),
- Vector w a)
- => (Int -> Int -> a)
- -> Mat v w a
-construct lambda = rows
+construct :: forall m n a. (Arity m, Arity n)
+ => (Int -> Int -> a) -> Mat m n a
+construct lambda = Mat $ generate make_row
where
- -- The arity trick is used in Data.Vector.Fixed.length.
- imax = (arity (undefined :: Dim v)) - 1
- jmax = (arity (undefined :: Dim w)) - 1
- row' i = V.fromList [ lambda i j | j <- [0..jmax] ]
- rows = V.fromList [ row' i | i <- [0..imax] ]
+ make_row :: Int -> Vec n a
+ make_row i = generate (lambda i)
+
+
+-- | Create an identity matrix with the right dimensions.
+--
+-- Examples:
+--
+-- >>> identity_matrix :: Mat3 Int
+-- ((1,0,0),(0,1,0),(0,0,1))
+-- >>> identity_matrix :: Mat3 Double
+-- ((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))
+--
+identity_matrix :: (Arity m, Ring.C a) => Mat m m a
+identity_matrix =
+ construct (\i j -> if i == j then (fromInteger 1) else (fromInteger 0))
-- | Given a positive-definite matrix @m@, computes the
-- upper-triangular matrix @r@ with (transpose r)*r == m and all
-- >>> let m1 = fromList [[20,-1], [-1,20]] :: Mat2 Double
-- >>> cholesky m1
-- ((4.47213595499958,-0.22360679774997896),(0.0,4.466542286825459))
--- >>> (transpose (cholesky m1)) `mult` (cholesky m1)
+-- >>> (transpose (cholesky m1)) * (cholesky m1)
-- ((20.000000000000004,-1.0),(-1.0,20.0))
--
-cholesky :: forall a v w.
- (RealFloat a,
- Vector v (Vn w a),
- Vector w a)
- => (Mat v w a)
- -> (Mat v w a)
+cholesky :: forall m n a. (Algebraic.C a, Arity m, Arity n)
+ => (Mat m n a) -> (Mat m n 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)
+ (((m !!! (i,j)) - sum [(r k i) NP.* (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.
+
+-- | Returns True if the given matrix is upper-triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_upper_triangular m
+-- False
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_upper_triangular m
+-- True
+--
+is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n)
+ => Mat m n a -> Bool
+is_upper_triangular m =
+ and $ concat results
+ where
+ results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
+
+ test :: Int -> Int -> Bool
+ test i j
+ | i <= j = True
+ | otherwise = m !!! (i,j) == 0
+
+
+-- | Returns True if the given matrix is lower-triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_lower_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_lower_triangular m
+-- False
+--
+is_lower_triangular :: (Eq a,
+ Ring.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_lower_triangular = is_upper_triangular . transpose
+
+
+-- | Returns True if the given matrix is triangular, and False
+-- otherwise.
--
-- Examples:
--
--- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat Vec2D Vec3D Int
--- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat Vec3D Vec2D Int
--- >>> m1 `mult` m2
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> is_triangular m
+-- False
+--
+is_triangular :: (Eq a,
+ Ring.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_triangular m = is_upper_triangular m || is_lower_triangular m
+
+
+-- | Return the (i,j)th minor of m.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> minor m 0 0 :: Mat2 Int
+-- ((5,6),(8,9))
+-- >>> minor m 1 1 :: Mat2 Int
+-- ((1,3),(7,9))
+--
+minor :: (m ~ S r,
+ n ~ S t,
+ Arity r,
+ Arity t)
+ => Mat m n a
+ -> Int
+ -> Int
+ -> Mat r t a
+minor (Mat rows) i j = m
+ where
+ rows' = delete rows i
+ m = Mat $ V.map ((flip delete) j) rows'
+
+
+class (Eq a, Ring.C a) => Determined p a where
+ determinant :: (p a) -> a
+
+instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
+ determinant (Mat rows) = (V.head . V.head) rows
+
+instance (Eq a,
+ Ring.C a,
+ Arity n,
+ Determined (Mat (S n) (S n)) a)
+ => Determined (Mat (S (S n)) (S (S n))) a where
+ -- | The recursive definition with a special-case for triangular matrices.
+ --
+ -- Examples:
+ --
+ -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> determinant m
+ -- -1
+ --
+ determinant m
+ | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
+ | otherwise = determinant_recursive
+ where
+ m' i j = m !!! (i,j)
+
+ det_minor i j = determinant (minor m i j)
+
+ determinant_recursive =
+ sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 0 j)
+ | j <- [0..(ncols m)-1] ]
+
+
+
+-- | Matrix multiplication.
+--
+-- Examples:
+--
+-- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat N2 N3 Int
+-- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat N3 N2 Int
+-- >>> m1 * m2
-- ((22,28),(49,64))
--
-mult :: (Num a,
- Vector v (Vn w a),
- Vector w a,
- Vector w (Vn z a),
- Vector z a,
- Vector v (Vn z a))
- => Mat v w a
- -> Mat w z a
- -> Mat v z a
-mult m1 m2 = construct lambda
+infixl 7 *
+(*) :: (Ring.C a, Arity m, Arity n, Arity p)
+ => Mat m n a
+ -> Mat n p a
+ -> Mat m p a
+(*) m1 m2 = construct lambda
where
lambda i j =
- sum [(m1 !!! (i,k)) * (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
+ sum [(m1 !!! (i,k)) NP.* (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
+
+
+
+instance (Ring.C a, Arity m, Arity n) => Additive.C (Mat m n 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, Arity m, Arity n, m ~ n) => Ring.C (Mat m n a) where
+ -- The first * is ring multiplication, the second is matrix
+ -- multiplication.
+ m1 * m2 = m1 * m2
+
+
+instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n a) where
+ -- We can multiply a matrix by a scalar of the same type as its
+ -- elements.
+ x *> (Mat rows) = Mat $ V.map (V.map (NP.* x)) rows
+
+
+instance (Algebraic.C a,
+ ToRational.C a,
+ Arity m)
+ => Normed (Mat (S m) N1 a) where
+ -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
+ -- all matrices as big vectors.
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = vec2d (3,4)
+ -- >>> norm_p 1 v1
+ -- 7.0
+ -- >>> norm_p 2 v1
+ -- 5.0
+ --
+ norm_p p (Mat rows) =
+ (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
+ where
+ p' = toInteger p
+ xs = concat $ V.toList $ V.map V.toList rows
+
+ -- | The infinity norm.
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = vec3d (1,5,2)
+ -- >>> norm_infty v1
+ -- 5
+ --
+ norm_infty (Mat rows) =
+ fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
+
+
+
+
+
+-- Vector helpers. We want it to be easy to create low-dimension
+-- column vectors, which are nx1 matrices.
+
+-- | Convenient constructor for 2D vectors.
+--
+-- Examples:
+--
+-- >>> import Roots.Simple
+-- >>> let fst m = m !!! (0,0)
+-- >>> let snd m = m !!! (1,0)
+-- >>> let h = 0.5 :: Double
+-- >>> let g1 m = 1.0 + h NP.* exp(-((fst m)^2))/(1.0 + (snd m)^2)
+-- >>> let g2 m = 0.5 + h NP.* atan((fst m)^2 + (snd m)^2)
+-- >>> let g u = vec2d ((g1 u), (g2 u))
+-- >>> let u0 = vec2d (1.0, 1.0)
+-- >>> let eps = 1/(10^9)
+-- >>> fixed_point g eps u0
+-- ((1.0728549599342185),(1.0820591495686167))
+--
+vec1d :: (a) -> Mat N1 N1 a
+vec1d (x) = Mat (mk1 (mk1 x))
+
+vec2d :: (a,a) -> Mat N2 N1 a
+vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
+
+vec3d :: (a,a,a) -> Mat N3 N1 a
+vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
+
+vec4d :: (a,a,a,a) -> Mat N4 N1 a
+vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
+
+vec5d :: (a,a,a,a,a) -> Mat N5 N1 a
+vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z))
+
+-- Since we commandeered multiplication, we need to create 1x1
+-- matrices in order to multiply things.
+scalar :: a -> Mat N1 N1 a
+scalar x = Mat (mk1 (mk1 x))
+
+dot :: (RealRing.C a, n ~ N1, m ~ S t, Arity t)
+ => Mat m n a
+ -> Mat m n a
+ -> a
+v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
+
+
+-- | The angle between @v1@ and @v2@ in Euclidean space.
+--
+-- Examples:
+--
+-- >>> let v1 = vec2d (1.0, 0.0)
+-- >>> let v2 = vec2d (0.0, 1.0)
+-- >>> angle v1 v2 == pi/2.0
+-- True
+--
+angle :: (Transcendental.C a,
+ RealRing.C a,
+ n ~ N1,
+ m ~ S t,
+ Arity t,
+ ToRational.C a)
+ => Mat m n a
+ -> Mat m n a
+ -> a
+angle v1 v2 =
+ acos theta
+ where
+ theta = (recip norms) NP.* (v1 `dot` v2)
+ norms = (norm v1) NP.* (norm v2)
+
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the on-diagonal entries of @matrix@. The
+-- off-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> diagonal_part m
+-- ((1,0,0),(0,5,0),(0,0,9))
+--
+diagonal_part :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+diagonal_part matrix =
+ construct lambda
+ where
+ lambda i j = if i == j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the on-diagonal and below-diagonal entries of
+-- @matrix@. The above-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> lt_part m
+-- ((1,0,0),(4,5,0),(7,8,9))
+--
+lt_part :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+lt_part matrix =
+ construct lambda
+ where
+ lambda i j = if i >= j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the below-diagonal entries of @matrix@. The on-
+-- and above-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> lt_part_strict m
+-- ((0,0,0),(4,0,0),(7,8,0))
+--
+lt_part_strict :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+lt_part_strict matrix =
+ construct lambda
+ where
+ lambda i j = if i > j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the on-diagonal and above-diagonal entries of
+-- @matrix@. The below-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ut_part m
+-- ((1,2,3),(0,5,6),(0,0,9))
+--
+ut_part :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+ut_part = transpose . lt_part . transpose
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the above-diagonal entries of @matrix@. The on-
+-- and below-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ut_part_strict m
+-- ((0,2,3),(0,0,6),(0,0,0))
+--
+ut_part_strict :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+ut_part_strict = transpose . lt_part_strict . transpose
projects / numerical-analysis.git / blobdiff