X-Git-Url: http://gitweb.michael.orlitzky.com/?p=numerical-analysis.git;a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=82665578cf037def7c04ef223a8b6e350ad9232f;hp=3d4daab3dc1f7be81277238b52253c5df54ae61f;hb=ca021dad591f47dbe1581c19c4ae4bf1fee821b9;hpb=cd3f0ee19d0894b6ab3b7ecc4e1045d7728e5bcc

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 3d4daab..8266557 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,
@@ -37,6 +30,7 @@ import qualified Data.Vector.Fixed as V (
   fromList,
   head,
   ifoldl,
+  ifoldr,
   imap,
   map,
   maximum,
@@ -46,6 +40,7 @@ import qualified Data.Vector.Fixed as V (
   zipWith )
 import Data.Vector.Fixed.Cont ( Arity, arity )
 import Linear.Vector ( Vec, delete, element_sum )
+import Naturals ( N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, S, Z )
 import Normed ( Normed(..) )
 
 import NumericPrelude hiding ( (*), abs )
@@ -55,6 +50,7 @@ import Algebra.Absolute ( abs )
 import qualified Algebra.Additive as Additive ( C )
 import qualified Algebra.Algebraic as Algebraic ( C )
 import Algebra.Algebraic ( root )
+import qualified Algebra.Field as Field ( C )
 import qualified Algebra.Ring as Ring ( C )
 import qualified Algebra.Module as Module ( C )
 import qualified Algebra.RealRing as RealRing ( C )
@@ -72,29 +68,32 @@ type Mat3 a = Mat N3 N3 a
 type Mat4 a = Mat N4 N4 a
 type Mat5 a = Mat N5 N5 a
 
+-- * Type synonyms for 1-by-n row "vectors".
+
 -- | Type synonym for row vectors expressed as 1-by-n matrices.
 type Row n a = Mat N1 n a
 
--- Type synonyms for 1-by-n row "vectors".
 type Row1 a = Row N1 a
 type Row2 a = Row N2 a
 type Row3 a = Row N3 a
 type Row4 a = Row N4 a
 type Row5 a = Row N5 a
 
+-- * Type synonyms for n-by-1 column "vectors".
+
 -- | Type synonym for column vectors expressed as n-by-1 matrices.
 type Col n a = Mat n N1 a
 
--- Type synonyms for n-by-1 column "vectors".
 type Col1 a = Col N1 a
 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
@@ -303,7 +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]
@@ -466,35 +464,50 @@ is_triangular :: (Ord a,
 is_triangular m = is_upper_triangular m || is_lower_triangular m
 
 
--- | Return the (i,j)th minor of m.
+-- | Delete the @i@th row and @j@th column from the matrix. The name
+--   \"preminor\" is made up, but is meant to signify that this is
+--   usually used in the computationof a minor. A minor is simply the
+--   determinant of a preminor in that case.
 --
 --   Examples:
 --
 --   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
---   >>> minor m 0 0 :: Mat2 Int
+--   >>> preminor m 0 0 :: Mat2 Int
 --   ((5,6),(8,9))
---   >>> minor m 1 1 :: Mat2 Int
+--   >>> preminor m 1 1 :: Mat2 Int
 --   ((1,3),(7,9))
 --
-minor :: (m ~ S r,
-          n ~ S t,
-          Arity r,
-          Arity t)
-      => Mat m n a
+preminor :: (Arity m, Arity n)
+      => Mat (S m) (S n) a
       -> Int
       -> Int
-      -> Mat r t a
-minor (Mat rows) i j = m
+      -> Mat m n a
+preminor (Mat rows) i j = m
   where
     rows' = delete rows i
     m = Mat $ V.map ((flip delete) j) rows'
 
 
+-- | Compute the i,jth minor of a @matrix@.
+--
+--   Examples:
+--
+--   >>> let m1 = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Double
+--   >>> minor m1 1 1
+--   -12.0
+--
+minor :: (Arity m, Determined (Mat m m) a)
+      => Mat (S m) (S m) a
+      -> Int
+      -> Int
+      -> a
+minor matrix i j = determinant (preminor matrix i j)
+
 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
+  determinant = unscalar
 
 instance (Ord a,
           Ring.C a,
@@ -516,10 +529,8 @@ instance (Ord a,
         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)
+            sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (minor m 0 j)
               | j <- [0..(ncols m)-1] ]
 
 
@@ -568,11 +579,12 @@ 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:
@@ -583,8 +595,12 @@ 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
@@ -657,16 +673,22 @@ 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
 scalar x = Mat (mk1 (mk1 x))
 
-dot :: (RealRing.C a, m ~ S t, Arity t)
-       => Col m a
-       -> Col m a
+-- Get the scalar value out of a 1x1 matrix.
+unscalar :: Mat1 a -> a
+unscalar (Mat rows) = V.head $ V.head rows
+
+
+dot :: (Ring.C a, Arity m)
+       => Col (S m) a
+       -> Col (S m) a
        -> a
-v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
+v1 `dot` v2 = unscalar $ ((transpose v1) * v2)
 
 
 -- | The angle between @v1@ and @v2@ in Euclidean space.
@@ -834,7 +856,7 @@ 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
@@ -906,7 +928,8 @@ map2 f (Mat rows) =
 
 
 -- | Fold over the entire matrix passing the coordinates @i@ and @j@
---   (of the row/column) to the accumulation function.
+--   (of the row/column) to the accumulation function. The fold occurs
+--   from top-left to bottom-right.
 --
 --   Examples:
 --
@@ -933,6 +956,36 @@ ifoldl2 f initial (Mat rows) =
     row_function rowinit idx r = V.ifoldl (g idx) rowinit r
 
 
+-- | 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.
+--
+--   The order of the arguments in the supplied function are different
+--   from those in V.ifoldr; we keep them similar to ifoldl2.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> ifoldr2 (\i j cur _ -> cur + i + j) 0 m
+--   18
+--
+ifoldr2 :: forall a b m n.
+           (Int -> Int -> b -> a -> b)
+        -> b
+        -> Mat m n a
+        -> b
+ifoldr2 f initial (Mat rows) =
+  V.ifoldr row_function initial rows
+  where
+    -- | Swap the order of arguments in @f@ so that it agrees with the
+    --   @f@ passed to ifoldl2.
+    g :: Int -> Int -> a -> b -> b
+    g w x y z = f w x z y
+
+    row_function :: Int -> Vec n a -> b -> b
+    row_function idx r rowinit = V.ifoldr (g idx) rowinit r
+
+
 -- | Map a function over a matrix of any dimensions, passing the
 --   coordinates @i@ and @j@ to the function @f@.
 --
@@ -969,3 +1022,62 @@ reverse2 :: (Arity m, Arity n) => Mat m n a -> Mat m n a
 reverse2 (Mat rows) = Mat $ V.reverse $ V.map V.reverse rows
 
 
+-- | Unsafely set the (i,j) element of the given matrix.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> set_idx m (1,1) 17
+--   ((1,2,3),(4,17,6),(7,8,9))
+--
+set_idx :: forall m n a.
+           (Arity m, Arity n)
+        => Mat m n a
+        -> (Int, Int)
+        -> a
+        -> Mat m n a
+set_idx matrix (i,j) newval =
+  imap2 updater matrix
+  where
+    updater :: Int -> Int -> a -> a
+    updater k l existing =
+      if k == i && l == j
+      then newval
+      else existing
+
+
+-- | Compute the i,jth cofactor of the given @matrix@. This simply
+--   premultiplues the i,jth minor by (-1)^(i+j).
+cofactor :: (Arity m, Determined (Mat m m) a)
+         => Mat (S m) (S m) a
+         -> Int
+         -> Int
+         -> a
+cofactor matrix i j =
+  (-1)^(toInteger i + toInteger j) NP.* (minor matrix i j)
+
+
+-- | Compute the inverse of a matrix using cofactor expansion
+--   (generalized Cramer's rule).
+--
+--   Examples:
+--
+--   >>> let m1 = fromList [[37,22],[17,54]] :: Mat2 Double
+--   >>> let e1 = [54/1624, -22/1624] :: [Double]
+--   >>> let e2 = [-17/1624, 37/1624] :: [Double]
+--   >>> let expected = fromList [e1, e2] :: Mat2 Double
+--   >>> let actual = inverse m1
+--   >>> frobenius_norm (actual - expected) < 1e-12
+--   True
+--
+inverse :: (Arity m,
+            Determined (Mat (S m) (S m)) a,
+            Determined (Mat m m) a,
+            Field.C a)
+        => Mat (S m) (S m) a
+        -> Mat (S m) (S m) a
+inverse matrix =
+  (1 / (determinant matrix)) *> (transpose $ construct lambda)
+  where
+    lambda i j = cofactor matrix i j
+