X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=3d4daab3dc1f7be81277238b52253c5df54ae61f;hb=cd3f0ee19d0894b6ab3b7ecc4e1045d7728e5bcc;hp=cfa838042ee7bf0794e7a5275981f7ab16ffd7fb;hpb=9fa2506d30661e984ff74313c3884007067387cb;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index cfa8380..3d4daab 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -31,8 +31,7 @@ import Data.Vector.Fixed (
   mk2,
   mk3,
   mk4,
-  mk5
-  )
+  mk5 )
 import qualified Data.Vector.Fixed as V (
   and,
   fromList,
@@ -42,6 +41,7 @@ import qualified Data.Vector.Fixed as V (
   map,
   maximum,
   replicate,
+  reverse,
   toList,
   zipWith )
 import Data.Vector.Fixed.Cont ( Arity, arity )
@@ -297,10 +297,34 @@ identity_matrix =
 --   Examples:
 --
 --   >>> let m1 = fromList [[20,-1], [-1,20]] :: Mat2 Double
---   >>> cholesky m1
---   ((4.47213595499958,-0.22360679774997896),(0.0,4.466542286825459))
---   >>> (transpose (cholesky m1)) * (cholesky m1)
---   ((20.000000000000004,-1.0),(-1.0,20.0))
+--   >>> let r = cholesky m1
+--   >>> frobenius_norm ((transpose r)*r - m1) < 1e-10
+--   True
+--   >>> 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]
+--   >>> let k4 = [0, 0, 0, 6, 0, 0, 0] :: [Double]
+--   >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
+--   >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
+--   >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
+--   >>> let big_K = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
+--
+--   >>> let e1 = [2.449489742783178,0,0,0,0,0,0] :: [Double]
+--   >>> let e2 = [-1.224744871391589,3,0,0,0,0,0] :: [Double]
+--   >>> let e3 = [0,-5/2,5/2,0,0,0,0] :: [Double]
+--   >>> let e4 = [0,0,0,2.449489742783178,0,0,0] :: [Double]
+--   >>> let e5 = [0,0,0,0,2.449489742783178,0,0] :: [Double]
+--   >>> let e6 = [0,0,0,0,0,2.449489742783178,0] :: [Double]
+--   >>> let e7 = [0,0,0,0,0,0,3.872983346207417] :: [Double]
+--   >>> let expected = fromList [e1,e2,e3,e4,e5,e6,e7] :: Mat N7 N7 Double
+--
+--   >>> let r = cholesky big_K
+--   >>> 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)
@@ -325,29 +349,26 @@ cholesky m = construct r
 --   >>> is_upper_triangular' 1e-10 m
 --   True
 --
---   TODO:
---
---     1. Don't cheat with lists.
---
-is_upper_triangular' :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+is_upper_triangular' :: forall m n a.
+                        (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
                     => a -- ^ The tolerance @epsilon@.
                     -> Mat m n a
                     -> Bool
-is_upper_triangular' epsilon m =
-  and $ concat results
+is_upper_triangular' epsilon matrix =
+  ifoldl2 f True matrix
   where
-    results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
-
-    test :: Int -> Int -> Bool
-    test i j
+    f :: Int -> Int -> Bool -> a -> Bool
+    f _ _ False _ = False
+    f i j True x
       | i <= j = True
       -- use "less than or equal to" so zero is a valid epsilon
-      | otherwise = abs (m !!! (i,j)) <= epsilon
+      | otherwise = abs x <= epsilon
 
 
 -- | Returns True if the given matrix is upper-triangular, and False
---   otherwise. A specialized version of 'is_upper_triangular\'' with
---   @epsilon = 0@.
+--   otherwise. We don't delegate to the general
+--   'is_upper_triangular'' here because it imposes additional
+--   typeclass constraints throughout the library.
 --
 --   Examples:
 --
@@ -359,18 +380,22 @@ is_upper_triangular' epsilon m =
 --   >>> is_upper_triangular m
 --   True
 --
---   TODO:
---
---     1. The Ord constraint is too strong here, Eq would suffice.
---
-is_upper_triangular :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+is_upper_triangular :: forall m n a.
+                       (Eq a, Ring.C a, Arity m, Arity n)
                     => Mat m n a -> Bool
-is_upper_triangular = is_upper_triangular' 0
+is_upper_triangular matrix =
+  ifoldl2 f True matrix
+  where
+    f :: Int -> Int -> Bool -> a -> Bool
+    f _ _ False _ = False
+    f i j True x
+      | i <= j = True
+      | otherwise = x == 0
+
 
 
 -- | Returns True if the given matrix is lower-triangular, and False
---   otherwise. This is a specialized version of 'is_lower_triangular\''
---   with @epsilon = 0@.
+--   otherwise.
 --
 --   Examples:
 --
@@ -382,9 +407,8 @@ is_upper_triangular = is_upper_triangular' 0
 --   >>> is_lower_triangular m
 --   False
 --
-is_lower_triangular :: (Ord a,
+is_lower_triangular :: (Eq a,
                         Ring.C a,
-                        Absolute.C a,
                         Arity m,
                         Arity n)
                     => Mat m n a
@@ -638,9 +662,9 @@ vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z))
 scalar :: a -> Mat1 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
+dot :: (RealRing.C a, m ~ S t, Arity t)
+       => Col m a
+       -> Col m a
        -> a
 v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
 
@@ -656,12 +680,11 @@ v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
 --
 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
+          => Col m a
+          -> Col m a
           -> a
 angle v1 v2 =
   acos theta
@@ -795,37 +818,73 @@ trace matrix =
     element_sum $ V.map V.head rows
 
 
--- | Zip together two column matrices.
+-- | Zip together two matrices.
+--
+--   TODO: don't cheat with construct (map V.zips instead).
 --
 --   Examples:
 --
 --   >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
 --   >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
---   >>> colzip m1 m2
+--   >>> zip2 m1 m2
 --   (((1,1)),((1,2)),((1,3)))
 --
-colzip :: Arity m => Col m a -> Col m a -> Col m (a,a)
-colzip c1 c2 =
+--   >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+--   >>> let m2 = fromList [[1,1],[1,1]] :: Mat2 Int
+--   >>> 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 m1 m2 =
   construct lambda
   where
-    lambda i j = (c1 !!! (i,j), c2 !!! (i,j))
+    lambda i j = (m1 !!! (i,j), m2 !!! (i,j))
 
 
--- | Zip together two column matrices using the supplied function.
+-- | Zip together three matrices.
+--
+--   TODO: don't cheat with construct (map V.zips instead).
+--
+--   Examples:
+--
+--   >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
+--   >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
+--   >>> let m3 = fromList [[4],[5],[6]] :: Col3 Int
+--   >>> zip2three m1 m2 m3
+--   (((1,1,4)),((1,2,5)),((1,3,6)))
+--
+--   >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+--   >>> let m2 = fromList [[1,1],[1,1]] :: Mat2 Int
+--   >>> let m3 = fromList [[8,2],[6,3]] :: Mat2 Int
+--   >>> zip2three m1 m2 m3
+--   (((1,1,8),(2,1,2)),((3,1,6),(4,1,3)))
+--
+zip2three :: (Arity m, Arity n)
+          => Mat m n a
+          -> Mat m n a
+          -> Mat m n a
+          -> Mat m n (a,a,a)
+zip2three m1 m2 m3 =
+  construct lambda
+  where
+    lambda i j = (m1 !!! (i,j), m2 !!! (i,j), m3 !!! (i,j))
+
+
+-- | Zip together two matrices using the supplied function.
 --
 --   Examples:
 --
 --   >>> let c1 = fromList [[1],[2],[3]] :: Col3 Integer
 --   >>> let c2 = fromList [[4],[5],[6]] :: Col3 Integer
---   >>> colzipwith (^) c1 c2
+--   >>> zipwith2 (^) c1 c2
 --   ((1),(32),(729))
 --
-colzipwith :: Arity m
+zipwith2 :: Arity m
            => (a -> a -> b)
            -> Col m a
            -> Col m a
            -> Col m b
-colzipwith f c1 c2 =
+zipwith2 f c1 c2 =
   construct lambda
   where
     lambda i j = f (c1 !!! (i,j)) (c2 !!! (i,j))
@@ -852,7 +911,7 @@ map2 f (Mat rows) =
 --   Examples:
 --
 --   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
---   >>>  ifoldl2 (\i j cur _ -> cur + i + j) 0 m
+--   >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
 --   18
 --
 ifoldl2 :: forall a b m n.
@@ -888,3 +947,25 @@ imap2 f (Mat rows) =
   Mat $ V.imap g rows
   where
     g i = V.imap (f i)
+
+
+-- | Reverse the order of elements in a matrix.
+--
+--   Examples:
+--
+--   >>> let m1 = fromList [[1,2,3]] :: Row3 Int
+--   >>> reverse2 m1
+--   ((3,2,1))
+--
+--   >>> let m1 = vec3d (1,2,3 :: Int)
+--   >>> reverse2 m1
+--   ((3),(2),(1))
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> reverse2 m
+--   ((9,8,7),(6,5,4),(3,2,1))
+--
+reverse2 :: (Arity m, Arity n) => Mat m n a -> Mat m n a
+reverse2 (Mat rows) = Mat $ V.reverse $ V.map V.reverse rows
+
+