X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=5bb5519664c5fa4028d284775493bb7da4f5598e;hb=221805e711475880d37fb7bb6d7a0b04689b1ebe;hp=63c0348cd9e6e0ce676ce5c17c1629890836a6e6;hpb=3c5015c938c96f70b15c6292198a01390ee6540a;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 63c0348..5bb5519 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -1,78 +1,141 @@
-{-# 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,
+  N1,
   Vector
   )
 import qualified Data.Vector.Fixed as V (
+  and,
   fromList,
   length,
   map,
-  toList
+  replicate,
+  toList,
+  zipWith
   )
-import Data.Vector.Fixed.Internal (arity)
-
+import Data.Vector.Fixed.Internal (Arity, arity, S)
 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 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 (!)
 
@@ -89,16 +152,16 @@ column m j =
 --   >>> 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:
@@ -111,7 +174,7 @@ transpose m = V.fromList column_list
 --   >>> symmetric m2
 --   False
 --
-symmetric :: (Vector v (Vn w a),
+symmetric :: (Vector v (w a),
               Vector w a,
               v ~ w,
               Vector w Bool,
@@ -136,11 +199,11 @@ symmetric m =
 --   ((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
@@ -157,43 +220,333 @@ construct lambda = rows
 --   >>> 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)
+            (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)
+              (((m !!! (i,j)) - sum [(r k i) NP.* (r k j) | k <- [0..i-1]]))/(r i i)
           | otherwise = 0
 
+
+-- | 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, Vector w a) => Mat v w 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,
+                        Vector w a,
+                        Vector w (v a),
+                        Vector v a)
+                    => Mat v w a
+                    -> Bool
+is_lower_triangular = is_upper_triangular . transpose
+
+
+-- | Returns True if the given matrix is triangular, and False
+--   otherwise.
+--
+--   Examples:
+--
+--   >>> 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,
+                  Vector w a,
+                  Vector w (v a),
+                  Vector v a)
+               => Mat v w 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 :: (Dim v ~ S (Dim u),
+          Dim w ~ S (Dim z),
+          Vector z a,
+          Vector u (w a),
+          Vector u (z a))
+      => Mat v w a
+      -> Int
+      -> Int
+      -> Mat u z a
+minor (Mat rows) i j = m
+  where
+    rows' = delete rows i
+    m = Mat $ V.map ((flip delete) j) rows'
+
+
+determinant :: (Eq a,
+                Ring.C a,
+                Vector w a,
+                Vector w (v a),
+                Vector v a,
+                Dim v ~ S r,
+                Dim w ~ S t)
+             => Mat v w a
+             -> a
+determinant m
+  | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
+  | otherwise = undefined --determinant_recursive m
+
+{-
+determinant_recursive :: forall v w a r c.
+                         (Eq a,
+                          Ring.C a,
+                          Vector w a)
+                       => Mat (v r) (w c) a
+                       -> a
+determinant_recursive m
+  | (ncols m) == 0 || (nrows m) == 0 = error "don't do that"
+  | (ncols m) == 1 && (nrows m) == 1 = m !!! (0,0) -- Base case
+  | otherwise =
+      sum [ (-1)^(1+(toInteger j)) NP.* (m' 1 j) NP.* (det_minor 1 j)
+            | j <- [0..(ncols m)-1] ]
+      where
+        m' i j = m !!! (i,j)
+
+        det_minor :: Int -> Int -> a
+        det_minor i j = determinant (minor m i j)
+-}
+
 -- | Matrix multiplication. Our 'Num' instance doesn't define one, and
 --   we need additional restrictions on the result type anyway.
 --
 --   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 m1 = fromList [[1,2,3], [4,5,6]]  :: Mat D2 D3 Int
+--   >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat D3 D2 Int
+--   >>> m1 * m2
 --   ((22,28),(49,64))
 --
-mult :: (Num a,
-         Vector v (Vn w a),
+infixl 7 *
+(*) :: (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
-mult m1 m2 = construct lambda
+(*) 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,
+          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
+  -- The first * is ring multiplication, the second is matrix
+  -- multiplication.
+  m1 * m2 = m1 * m2
+
+
+instance (Ring.C a,
+          Vector v (w a),
+          Vector w a)
+         => Module.C a (Mat v w 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,
+          Vector v (w a),
+          Vector w a,
+          Vector v a,
+          Vector v [a])
+         => Normed (Mat v w 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. 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 = vec3d (1,5,2)
+  --   >>> norm_infty v1
+  --   5
+  --
+  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, which are nx1 matrices.
+
+-- | Convenient constructor for 2D vectors.
+--
+--   Examples:
+--
+--   >>> import Roots.Simple
+--   >>> let h = 0.5 :: Double
+--   >>> let g1 (Mat (D2 (D1 x) (D1 y))) = 1.0 + h NP.* exp(-(x^2))/(1.0 + y^2)
+--   >>> let g2 (Mat (D2 (D1 x) (D1 y))) = 0.5 + h NP.* atan(x^2 + y^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 D1 D1 a
+vec1d (x) = Mat (D1 (D1 x))
+
+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))
+
+-- Since we commandeered multiplication, we need to create 1x1
+-- matrices in order to multiply things.
+scalar :: a -> Mat D1 D1 a
+scalar x = Mat (D1 (D1 x))
+
+dot :: (RealRing.C a,
+        Dim w ~ N1,
+        Dim v ~ S n,
+        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) * 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,
+          Dim w ~ N1,
+          Dim v ~ S n,
+          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) NP.* (v1 `dot` v2)
+   norms = (norm v1) NP.* (norm v2)