X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=8006bdea1c2ee624930aa674e980635246c81d5a;hb=6892f38cf4dab275a91fbe06bfb8efe229518336;hp=2fe53401587f1fe25ec4ebc16919d1463c543691;hpb=cb41ccadccd4305065c3576d63d505a5d35c5279;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 2fe5340..8006bde 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -19,13 +19,6 @@ import Data.List (intercalate)
 
 import Data.Vector.Fixed (
   (!),
-  N1,
-  N2,
-  N3,
-  N4,
-  N5,
-  S,
-  Z,
   generate,
   mk1,
   mk2,
@@ -46,7 +39,8 @@ import qualified Data.Vector.Fixed as V (
   toList,
   zipWith )
 import Data.Vector.Fixed.Cont ( Arity, arity )
-import Linear.Vector ( Vec, delete, element_sum )
+import Linear.Vector ( Vec, delete )
+import Naturals ( N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, S, Z )
 import Normed ( Normed(..) )
 
 import NumericPrelude hiding ( (*), abs )
@@ -95,10 +89,11 @@ type Col2 a = Col N2 a
 type Col3 a = Col N3 a
 type Col4 a = Col N4 a
 type Col5 a = Col N5 a
-
--- We need a big column for Gaussian quadrature.
-type N10 = S (S (S (S (S N5))))
-type Col10 a = Col N10 a
+type Col6 a = Col N6 a
+type Col7 a = Col N7 a
+type Col8 a = Col N8 a
+type Col9 a = Col N9 a
+type Col10 a = Col N10 a -- We need a big column for Gaussian quadrature.
 
 
 instance (Eq a) => Eq (Mat m n a) where
@@ -307,7 +302,6 @@ identity_matrix =
 --   >>> is_upper_triangular r
 --   True
 --
---   >>> import Naturals ( N7 )
 --   >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
 --   >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
 --   >>> let k3 = [0, -7.5, 12.5, 0, 0, 0, 0] :: [Double]
@@ -552,13 +546,12 @@ instance (Ord a,
 --
 infixl 7 *
 (*) :: (Ring.C a, Arity m, Arity n, Arity p)
-        => Mat m n a
-        -> Mat n p a
-        -> Mat m p a
+        => Mat (S m) (S n) a
+        -> Mat (S n) (S p) a
+        -> Mat (S m) (S p) a
 (*) m1 m2 = construct lambda
   where
-    lambda i j =
-      sum [(m1 !!! (i,k)) NP.* (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
+    lambda i j = (transpose $ row m1 i) `dot` (column m2 j)
 
 
 
@@ -573,7 +566,7 @@ instance (Ring.C a, Arity m, Arity n) => Additive.C (Mat m n a) where
   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
+instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat (S m) (S n) a) where
   -- The first * is ring multiplication, the second is matrix
   -- multiplication.
   m1 * m2 = m1 * m2
@@ -585,11 +578,12 @@ instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n a) where
   x *> (Mat rows) = Mat $ V.map (V.map (NP.* x)) rows
 
 
-instance (Algebraic.C a,
+instance (Absolute.C a,
+          Algebraic.C a,
           ToRational.C a,
           Arity m)
-         => Normed (Mat (S m) N1 a) where
-  -- | Generic p-norms for vectors in R^n that are represented as nx1
+         => Normed (Col (S m) a) where
+  -- | Generic p-norms for vectors in R^n that are represented as n-by-1
   --   matrices.
   --
   --   Examples:
@@ -600,8 +594,12 @@ instance (Algebraic.C a,
   --   >>> norm_p 2 v1
   --   5.0
   --
+  --   >>> let v1 = vec2d (-1,1) :: Col2 Double
+  --   >>> norm_p 1 v1 :: Double
+  --   2.0
+  --
   norm_p p (Mat rows) =
-    (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
+    (root p') $ sum [fromRational' (toRational $ abs x)^p' | x <- xs]
     where
       p' = toInteger p
       xs = concat $ V.toList $ V.map V.toList rows
@@ -632,12 +630,13 @@ instance (Algebraic.C a,
 --   >>> frobenius_norm m == 3
 --   True
 --
-frobenius_norm :: (Algebraic.C a, Ring.C a) => Mat m n a -> a
-frobenius_norm (Mat rows) =
-  sqrt $ element_sum $ V.map row_sum rows
+frobenius_norm :: (Arity m, Arity n, Algebraic.C a, Ring.C a)
+               => Mat m n a
+               -> a
+frobenius_norm matrix =
+  sqrt $ element_sum2 $ squares
   where
-    -- | Square and add up the entries of a row.
-    row_sum = element_sum . V.map (^2)
+    squares = map2 (^2) matrix
 
 
 -- Vector helpers. We want it to be easy to create low-dimension
@@ -674,6 +673,7 @@ vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
 vec5d :: (a,a,a,a,a) -> Col5 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 -> Mat1 a
@@ -688,7 +688,7 @@ dot :: (Ring.C a, Arity m)
        => Col (S m) a
        -> Col (S m) a
        -> a
-v1 `dot` v2 = unscalar $ ((transpose v1) * v2)
+v1 `dot` v2 = element_sum2 $ zipwith2 (NP.*) v1 v2
 
 
 -- | The angle between @v1@ and @v2@ in Euclidean space.
@@ -834,10 +834,8 @@ ut_part_strict = transpose . lt_part_strict . transpose
 --   15
 --
 trace :: (Arity m, Ring.C a) => Mat m m a -> a
-trace matrix =
-  let (Mat rows) = diagonal matrix
-  in
-    element_sum $ V.map V.head rows
+trace = element_sum2 . diagonal
+
 
 
 -- | Zip together two matrices.
@@ -856,7 +854,7 @@ trace matrix =
 --   >>> zip2 m1 m2
 --   (((1,1),(2,1)),((3,1),(4,1)))
 --
-zip2 :: (Arity m, Arity n) => Mat m n a -> Mat m n a -> Mat m n (a,a)
+zip2 :: (Arity m, Arity n) => Mat m n a -> Mat m n b -> Mat m n (a,b)
 zip2 m1 m2 =
   construct lambda
   where
@@ -901,11 +899,11 @@ zip2three m1 m2 m3 =
 --   >>> zipwith2 (^) c1 c2
 --   ((1),(32),(729))
 --
-zipwith2 :: Arity m
-           => (a -> a -> b)
-           -> Col m a
-           -> Col m a
-           -> Col m b
+zipwith2 :: (Arity m, Arity n)
+           => (a -> b -> c)
+           -> Mat m n a
+           -> Mat m n b
+           -> Mat m n c
 zipwith2 f c1 c2 =
   construct lambda
   where
@@ -956,6 +954,18 @@ ifoldl2 f initial (Mat rows) =
     row_function rowinit idx r = V.ifoldl (g idx) rowinit r
 
 
+-- | Left fold over the entries of a matrix (top-left to bottom-right).
+--
+foldl2 :: forall a b m n.
+          (b -> a -> b)
+        -> b
+        -> Mat m n a
+        -> b
+foldl2 f initial matrix =
+  -- Use the index fold but ignore the index arguments.
+  let g _ _ = f in ifoldl2 g initial matrix
+
+
 -- | Fold over the entire matrix passing the coordinates @i@ and @j@
 --   (of the row/column) to the accumulation function. The fold occurs
 --   from bottom-right to top-left.
@@ -1081,3 +1091,35 @@ inverse matrix =
   where
     lambda i j = cofactor matrix i j
 
+
+
+-- | Retrieve the rows of a matrix as a column matrix. If the given
+--   matrix is m-by-n, the result would be an m-by-1 column whose
+--   entries are 1-by-n row matrices.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+--   >>> (rows2 m) !!! (0,0)
+--   ((1,2))
+--   >>> (rows2 m) !!! (1,0)
+--   ((3,4))
+--
+rows2 :: (Arity m, Arity n)
+      => Mat m n a
+      -> Col m (Row n a)
+rows2 (Mat rows) =
+  Mat $ V.map (mk1. Mat . mk1) rows
+
+
+
+-- | Sum the elements of a matrix.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,-1],[3,4]] :: Mat2 Int
+--   >>> element_sum2 m
+--   7
+--
+element_sum2 :: (Arity m, Arity n, Additive.C a) => Mat m n a -> a
+element_sum2 = foldl2 (+) zero