X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=e8d7180fbc7717ef93ce3eb30ff55332aeef31b5;hb=3274b18a2fb48f26e6582d9dcee11b4067230911;hp=3d4daab3dc1f7be81277238b52253c5df54ae61f;hpb=cd3f0ee19d0894b6ab3b7ecc4e1045d7728e5bcc;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 3d4daab..e8d7180 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -19,14 +19,8 @@ import Data.List (intercalate)
 
 import Data.Vector.Fixed (
   (!),
-  N1,
-  N2,
-  N3,
-  N4,
-  N5,
-  S,
-  Z,
   generate,
+  mk0,
   mk1,
   mk2,
   mk3,
@@ -34,67 +28,101 @@ import Data.Vector.Fixed (
   mk5 )
 import qualified Data.Vector.Fixed as V (
   and,
+  foldl,
   fromList,
   head,
   ifoldl,
+  ifoldr,
   imap,
   map,
-  maximum,
   replicate,
   reverse,
   toList,
   zipWith )
 import Data.Vector.Fixed.Cont ( Arity, arity )
-import Linear.Vector ( Vec, delete, element_sum )
+import Linear.Vector ( Vec, delete )
+import Naturals
 import Normed ( Normed(..) )
 
-import NumericPrelude hiding ( (*), abs )
+-- We want the "max" that works on Ord, not the one that only works on
+-- Bool/Integer from the Lattice class!
+import NumericPrelude hiding ( (*), abs, max)
 import qualified NumericPrelude as NP ( (*) )
 import qualified Algebra.Absolute as Absolute ( C )
 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 )
 import qualified Algebra.ToRational as ToRational ( C )
 import qualified Algebra.Transcendental as Transcendental ( C )
-import qualified Prelude as P ( map )
+import qualified Prelude as P ( map, max)
 
 -- | Our main matrix type.
 data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
 
 -- Type synonyms for n-by-n matrices.
+type Mat0 a = Mat Z Z 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
+type Mat6 a = Mat N6 N6 a
+type Mat7 a = Mat N7 N7 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 Col0 a = Col Z a
 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 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
+type Col11 a = Col N11 a
+type Col12 a = Col N12 a
+type Col13 a = Col N13 a
+type Col14 a = Col N14 a
+type Col15 a = Col N15 a
+type Col16 a = Col N16 a
+type Col17 a = Col N17 a
+type Col18 a = Col N18 a
+type Col19 a = Col N19 a
+type Col20 a = Col N20 a
+type Col21 a = Col N21 a
+type Col22 a = Col N22 a
+type Col23 a = Col N23 a
+type Col24 a = Col N24 a
+type Col25 a = Col N25 a
+type Col26 a = Col N26 a
+type Col27 a = Col N27 a
+type Col28 a = Col N28 a
+type Col29 a = Col N29 a
+type Col30 a = Col N30 a
+type Col31 a = Col N31 a
+type Col32 a = Col N32 a
 
 
 instance (Eq a) => Eq (Mat m n a) where
@@ -110,8 +138,8 @@ instance (Eq a) => Eq (Mat m n a) where
   --   >>> m1 == m3
   --   False
   --
-  (Mat rows1) == (Mat rows2) =
-    V.and $ V.zipWith comp rows1 rows2
+  (Mat rows_one) == (Mat rows_two) =
+    V.and $ V.zipWith comp rows_one rows_two
     where
       -- Compare a row, one column at a time.
       comp row1 row2 = V.and (V.zipWith (==) row1 row2)
@@ -303,7 +331,6 @@ 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]
@@ -326,15 +353,21 @@ identity_matrix =
 --   >>> frobenius_norm (r - (transpose expected)) < 1e-12
 --   True
 --
-cholesky :: forall m n a. (Algebraic.C a, Arity m, Arity n)
-         => (Mat m n a) -> (Mat m n a)
-cholesky m = construct r
+cholesky :: forall m a. (Algebraic.C a, Arity m)
+         => (Mat m m a) -> (Mat m m a)
+cholesky m = ifoldl2 f zero m
   where
-    r :: Int -> Int -> a
-    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) NP.* (r k j) | k <- [0..i-1]]))/(r i i)
-          | otherwise = 0
+    f :: Int -> Int -> (Mat m m a) -> a -> (Mat m m a)
+    f i j cur_R _ = set_idx cur_R (i,j) (r cur_R i j)
+
+    r :: (Mat m m a) -> Int -> Int -> a
+    r cur_R i j
+      | i == j = sqrt(m !!! (i,j) - sum [(cur_R !!! (k,i))^2 | k <- [0..i-1]])
+      | i < j = (((m !!! (i,j))
+                  - sum [(cur_R !!! (k,i)) NP.* (cur_R !!! (k,j))
+                             | k <- [0..i-1]]))/(cur_R !!! (i,i))
+      | otherwise = 0
+
 
 
 -- | Returns True if the given matrix is upper-triangular, and False
@@ -466,35 +499,50 @@ 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 +564,8 @@ 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] ]
 
 
@@ -535,28 +581,27 @@ 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)
 
 
 
 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 rows_one) + (Mat rows_two) =
+    Mat $ V.zipWith (V.zipWith (+)) rows_one rows_two
 
-  (Mat rows1) - (Mat rows2) =
-    Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
+  (Mat rows_one) - (Mat rows_two) =
+    Mat $ V.zipWith (V.zipWith (-)) rows_one rows_two
 
   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
@@ -568,11 +613,12 @@ 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 m a) where
+  -- | Generic p-norms for vectors in R^m that are represented as m-by-1
   --   matrices.
   --
   --   Examples:
@@ -583,8 +629,16 @@ instance (Algebraic.C a,
   --   >>> norm_p 2 v1
   --   5.0
   --
+  --   >>> let v1 = vec2d (-1,1) :: Col2 Double
+  --   >>> norm_p 1 v1 :: Double
+  --   2.0
+  --
+  --   >>> let v1 = vec0d :: Col0 Double
+  --   >>> norm v1
+  --   0.0
+  --
   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
@@ -598,7 +652,8 @@ instance (Algebraic.C a,
   --   5
   --
   norm_infty (Mat rows) =
-    fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
+    fromRational' $ toRational
+                      $ (V.foldl P.max 0) $ V.map (V.foldl P.max 0) rows
 
 
 -- | Compute the Frobenius norm of a matrix. This essentially treats
@@ -615,12 +670,13 @@ 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
@@ -642,6 +698,9 @@ frobenius_norm (Mat rows) =
 --   >>> fixed_point g eps u0
 --   ((1.0728549599342185),(1.0820591495686167))
 --
+vec0d :: Col0 a
+vec0d = Mat mk0
+
 vec1d :: (a) -> Col1 a
 vec1d (x) = Mat (mk1 (mk1 x))
 
@@ -657,16 +716,22 @@ 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 = element_sum2 $ zipwith2 (NP.*) v1 v2
 
 
 -- | The angle between @v1@ and @v2@ in Euclidean space.
@@ -812,10 +877,8 @@ 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.
@@ -834,7 +897,7 @@ 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
@@ -879,11 +942,11 @@ 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
@@ -906,7 +969,8 @@ 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 +997,48 @@ 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.
+--
+--   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 +1075,94 @@ 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
+
+
+
+-- | 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