X-Git-Url: http://gitweb.michael.orlitzky.com/?p=numerical-analysis.git;a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=82665578cf037def7c04ef223a8b6e350ad9232f;hp=39576dc41343c2bbad77b745a33bc720d44f666f;hb=ca021dad591f47dbe1581c19c4ae4bf1fee821b9;hpb=303c5e7bba583f08e59bc6c848be8e75c1155a3b

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 39576dc..8266557 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -2,60 +2,101 @@
 {-# LANGUAGE FlexibleContexts #-}
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE NoMonomorphismRestriction #-}
 {-# LANGUAGE ScopedTypeVariables #-}
 {-# LANGUAGE TypeFamilies #-}
 {-# LANGUAGE RebindableSyntax #-}
 
+-- | Boxed matrices; that is, boxed m-vectors of boxed n-vectors. We
+--   assume that the underlying representation is
+--   Data.Vector.Fixed.Boxed.Vec for simplicity. It was tried in
+--   generality and failed.
+--
 module Linear.Matrix
 where
 
 import Data.List (intercalate)
 
 import Data.Vector.Fixed (
-  Dim,
-  Vector
-  )
+  (!),
+  generate,
+  mk1,
+  mk2,
+  mk3,
+  mk4,
+  mk5 )
 import qualified Data.Vector.Fixed as V (
-  Fun(..),
-  N1,
   and,
-  eq,
-  foldl,
   fromList,
-  length,
+  head,
+  ifoldl,
+  ifoldr,
+  imap,
   map,
   maximum,
   replicate,
+  reverse,
   toList,
-  zipWith
-  )
-import Data.Vector.Fixed.Internal (Arity, arity, S, Dim)
-import Linear.Vector
-import Normed
-
-import NumericPrelude hiding (abs)
-import qualified Algebra.Algebraic as Algebraic
-import qualified Algebra.Absolute as Absolute
-import qualified Algebra.Additive as Additive
-import qualified Algebra.Ring as Ring
-import Algebra.Absolute (abs)
-import qualified Algebra.Field as Field
-import qualified Algebra.RealField as RealField
-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
+  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 )
+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 )
+
+-- | 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 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 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 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 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
+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
   -- | Compare a row at a time.
   --
   --   Examples:
@@ -75,7 +116,7 @@ instance (Eq a, Vector v Bool, Vector w Bool) => Eq (Mat v w a) where
       comp row1 row2 = V.and (V.zipWith (==) row1 row2)
 
 
-instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where
+instance (Show a) => Show (Mat m n 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
@@ -99,55 +140,89 @@ instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where
           element_strings = P.map show v1l
 
 
-
 -- | Convert a matrix to a nested list.
-toList :: Mat v w a -> [[a]]
+toList :: Mat m n a -> [[a]]
 toList (Mat rows) = map V.toList (V.toList rows)
 
+
 -- | Create a matrix from a nested list.
-fromList :: (Vector v (w a), Vector w a, Vector v a) => [[a]] -> Mat v w a
+fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
 fromList vs = Mat (V.fromList $ map V.fromList vs)
 
 
--- | Unsafe indexing.
-(!!!) :: (Vector w a) => Mat v w a -> (Int, Int) -> a
-(!!!) m (i, j) = (row m i) ! j
+-- | Unsafe indexing. Much faster than the safe indexing.
+(!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
+(!!!) (Mat rows) (i, j) = (rows ! i) ! j
+
 
 -- | Safe indexing.
-(!!?) :: Mat v w a -> (Int, Int) -> Maybe a
-(!!?) m@(Mat rows) (i, j)
-  | i < 0 || j < 0 = Nothing
-  | i > V.length rows = Nothing
-  | otherwise = if j > V.length (row m j)
-                then Nothing
-                else Just $ (row m j) ! j
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+--   >>> m !!? (-1,-1)
+--   Nothing
+--   >>>  m !!? (-1,0)
+--   Nothing
+--   >>>  m !!? (-1,1)
+--   Nothing
+--   >>>  m !!? (0,-1)
+--   Nothing
+--   >>>  m !!? (0,0)
+--   Just 1
+--   >>>  m !!? (0,1)
+--   Just 2
+--   >>>  m !!? (1,-1)
+--   Nothing
+--   >>>  m !!? (1,0)
+--   Just 3
+--   >>>  m !!? (1,1)
+--   Just 4
+--   >>>  m !!? (2,-1)
+--   Nothing
+--   >>>  m !!? (2,0)
+--   Nothing
+--   >>>  m !!? (2,1)
+--   Nothing
+--   >>>  m !!? (2,2)
+--   Nothing
+--
+(!!?) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> Maybe a
+(!!?) matrix idx =
+  ifoldl2 f Nothing matrix
+  where
+    f k l found cur = if (k,l) == idx then (Just cur) else found
 
 
 -- | The number of rows in the matrix.
-nrows :: Mat v w a -> Int
-nrows (Mat rows) = V.length rows
+nrows :: forall m n a. (Arity m) => Mat m n a -> Int
+nrows _ = arity (undefined :: m)
+
 
 -- | The number of columns in the first row of the
 --   matrix. Implementation stolen from Data.Vector.Fixed.length.
-ncols :: forall v w a. (Vector w a) => Mat v w a -> Int
-ncols _ = (arity (undefined :: Dim w))
+ncols :: forall m n a. (Arity n) => Mat m n a -> Int
+ncols _ = arity (undefined :: n)
 
--- | Return the @i@th row of @m@. Unsafe.
-row :: Mat v w a -> Int -> w a
-row (Mat rows) i = rows ! i
+
+-- | Return the @i@th row of @m@ as a matrix. Unsafe.
+row :: (Arity m, Arity n) => Mat m n a -> Int -> Row n a
+row m i =
+  construct lambda
+  where
+    lambda _ j = m !!! (i, j)
 
 
--- | Return the @j@th column of @m@. Unsafe.
-column :: (Vector v a) => Mat v w a -> Int -> v a
-column (Mat rows) j =
-  V.map (element j) rows
+-- | Return the @j@th column of @m@ as a matrix. Unsafe.
+column :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
+column m j =
+  construct lambda
   where
-    element = flip (!)
+    lambda i _ = m !!! (i, j)
 
 
 -- | Transpose @m@; switch it's columns and its rows. This is a dirty
---   implementation.. it would be a little cleaner to use imap, but it
---   doesn't seem to work.
+--   implementation, but I don't see a better way.
 --
 --   TODO: Don't cheat with fromList.
 --
@@ -157,14 +232,11 @@ column (Mat rows) j =
 --   >>> transpose m
 --   ((1,3),(2,4))
 --
-transpose :: (Vector w (v a),
-              Vector v a,
-              Vector w a)
-             => Mat v w a
-             -> Mat w v a
-transpose m = Mat $ V.fromList column_list
+transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
+transpose matrix =
+  construct lambda
   where
-    column_list = [ column m i | i <- [0..(ncols m)-1] ]
+    lambda i j = matrix !!! (j,i)
 
 
 -- | Is @m@ symmetric?
@@ -179,13 +251,7 @@ transpose m = Mat $ V.fromList column_list
 --   >>> symmetric m2
 --   False
 --
-symmetric :: (Vector v (w a),
-              Vector w a,
-              v ~ w,
-              Vector w Bool,
-              Eq a)
-             => Mat v w a
-             -> Bool
+symmetric :: (Eq a, Arity m) => Mat m m a -> Bool
 symmetric m =
   m == (transpose m)
 
@@ -195,26 +261,33 @@ symmetric m =
 --   entries in the matrix. The i,j entry of the resulting matrix will
 --   have the value returned by lambda i j.
 --
---   TODO: Don't cheat with fromList.
---
 --   Examples:
 --
 --   >>> let lambda i j = i + j
 --   >>> construct lambda :: Mat3 Int
 --   ((0,1,2),(1,2,3),(2,3,4))
 --
-construct :: forall v w a.
-             (Vector v (w a),
-              Vector w a)
-             => (Int -> Int -> a)
-             -> Mat v w a
-construct lambda = Mat rows
+construct :: forall m n a. (Arity m, Arity n)
+          => (Int -> Int -> a) -> Mat m n a
+construct lambda = Mat $ generate make_row
   where
-    -- The arity trick is used in Data.Vector.Fixed.length.
-    imax = (arity (undefined :: Dim v)) - 1
-    jmax = (arity (undefined :: Dim w)) - 1
-    row' i = V.fromList [ lambda i j | j <- [0..jmax] ]
-    rows = V.fromList [ row' i | i <- [0..imax] ]
+    make_row :: Int -> Vec n a
+    make_row i = generate (lambda i)
+
+
+-- | Create an identity matrix with the right dimensions.
+--
+--   Examples:
+--
+--   >>> identity_matrix :: Mat3 Int
+--   ((1,0,0),(0,1,0),(0,0,1))
+--   >>> identity_matrix :: Mat3 Double
+--   ((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))
+--
+identity_matrix :: (Arity m, Ring.C a) => Mat m m a
+identity_matrix =
+  construct (\i j -> if i == j then (fromInteger 1) else (fromInteger 0))
+
 
 -- | Given a positive-definite matrix @m@, computes the
 --   upper-triangular matrix @r@ with (transpose r)*r == m and all
@@ -223,56 +296,267 @@ construct lambda = Mat rows
 --   Examples:
 --
 --   >>> 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)
---   ((20.000000000000004,-1.0),(-1.0,20.0))
---
-cholesky :: forall a v w.
-            (Algebraic.C a,
-             Vector v (w a),
-             Vector w a,
-             Vector v a)
-            => (Mat v w a)
-            -> (Mat v w a)
+--   >>> let r = cholesky m1
+--   >>> frobenius_norm ((transpose r)*r - m1) < 1e-10
+--   True
+--   >>> is_upper_triangular r
+--   True
+--
+--   >>> 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)
 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]])
           | 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
 
 
--- | Matrix multiplication. Our 'Num' instance doesn't define one, and
---   we need additional restrictions on the result type anyway.
+-- | Returns True if the given matrix is upper-triangular, and False
+--   otherwise. The parameter @epsilon@ lets the caller choose a
+--   tolerance.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
+--   >>> is_upper_triangular m
+--   False
+--   >>> is_upper_triangular' 1e-10 m
+--   True
+--
+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 matrix =
+  ifoldl2 f True matrix
+  where
+    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 x <= epsilon
+
+
+-- | Returns True if the given matrix is upper-triangular, and False
+--   otherwise. We don't delegate to the general
+--   'is_upper_triangular'' here because it imposes additional
+--   typeclass constraints throughout the library.
+--
+--   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 :: forall m n a.
+                       (Eq a, Ring.C a, Arity m, Arity n)
+                    => Mat m n a -> Bool
+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.
 --
 --   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 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,
+                        Arity m,
+                        Arity n)
+                    => Mat m n a
+                    -> Bool
+is_lower_triangular = is_upper_triangular . transpose
+
+
+-- | Returns True if the given matrix is lower-triangular, and False
+--   otherwise. The parameter @epsilon@ lets the caller choose a
+--   tolerance.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,1e-12],[1,1]] :: Mat2 Double
+--   >>> is_lower_triangular m
+--   False
+--   >>> is_lower_triangular' 1e-12 m
+--   True
+--
+is_lower_triangular' :: (Ord a,
+                         Ring.C a,
+                         Absolute.C a,
+                         Arity m,
+                         Arity n)
+                    => a -- ^ The tolerance @epsilon@.
+                    -> Mat m n a
+                    -> Bool
+is_lower_triangular' epsilon = (is_upper_triangular' epsilon) . 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 :: (Ord a,
+                  Ring.C a,
+                  Absolute.C a,
+                  Arity m,
+                  Arity n)
+              => Mat m n a
+              -> Bool
+is_triangular m = is_upper_triangular m || is_lower_triangular 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
+--   >>> preminor m 0 0 :: Mat2 Int
+--   ((5,6),(8,9))
+--   >>> preminor m 1 1 :: Mat2 Int
+--   ((1,3),(7,9))
+--
+preminor :: (Arity m, Arity n)
+      => Mat (S m) (S n) a
+      -> Int
+      -> Int
+      -> 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 = unscalar
+
+instance (Ord a,
+          Ring.C a,
+          Absolute.C a,
+          Arity n,
+          Determined (Mat (S n) (S n)) a)
+         => Determined (Mat (S (S n)) (S (S n))) a where
+  -- | The recursive definition with a special-case for triangular matrices.
+  --
+  --   Examples:
+  --
+  --   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> determinant m
+  --   -1
+  --
+  determinant m
+    | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
+    | otherwise = determinant_recursive
+        where
+          m' i j = m !!! (i,j)
+
+          determinant_recursive =
+            sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (minor m 0 j)
+              | j <- [0..(ncols m)-1] ]
+
+
+
+-- | Matrix multiplication.
+--
+--   Examples:
+--
+--   >>> let m1 = fromList [[1,2,3], [4,5,6]]  :: Mat N2 N3 Int
+--   >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat N3 N2 Int
+--   >>> m1 * m2
 --   ((22,28),(49,64))
 --
-mult :: (Ring.C a,
-         Vector v a,
-         Vector w a,
-         Vector z a,
-         Vector v (z a))
-        => Mat v w a
-        -> Mat w z a
-        -> Mat v z a
-mult m1 m2 = construct lambda
+infixl 7 *
+(*) :: (Ring.C a, Arity m, Arity n, Arity p)
+        => Mat m n a
+        -> Mat n p a
+        -> Mat m p a
+(*) 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
+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
@@ -283,87 +567,517 @@ instance (Ring.C a,
   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
-    one = Mat (V.replicate $ V.replicate (fromInteger 1))
-    m1 * m2 = m1 `mult` m2
+instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat m n a) where
+  -- The first * is ring multiplication, the second is matrix
+  -- multiplication.
+  m1 * m2 = m1 * m2
 
 
-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
-   -- Treat the matrix as a big vector.
-   norm_p p (Mat rows) =
-     sqrt $ sum [(fromRational' $ toRational x)^2 | x <- xs]
-     where
-       xs = concat $ V.toList $ V.map V.toList rows
+instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n 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
 
-   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
 
+instance (Absolute.C a,
+          Algebraic.C a,
+          ToRational.C a,
+          Arity m)
+         => Normed (Col (S m) a) where
+  -- | Generic p-norms for vectors in R^n that are represented as n-by-1
+  --   matrices.
+  --
+  --   Examples:
+  --
+  --   >>> let v1 = vec2d (3,4)
+  --   >>> norm_p 1 v1
+  --   7.0
+  --   >>> 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 $ abs x)^p' | x <- xs]
+    where
+      p' = toInteger p
+      xs = concat $ V.toList $ V.map V.toList rows
+
+  -- | The infinity norm.
+  --
+  --   Examples:
+  --
+  --   >>> let v1 = vec3d (1,5,2)
+  --   >>> norm_infty v1
+  --   5
+  --
+  norm_infty (Mat rows) =
+    fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
 
 
+-- | Compute the Frobenius norm of a matrix. This essentially treats
+--   the matrix as one long vector containing all of its entries (in
+--   any order, it doesn't matter).
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1, 2, 3],[4,5,6],[7,8,9]] :: Mat3 Double
+--   >>> frobenius_norm m == sqrt 285
+--   True
+--
+--   >>> let m = fromList [[1, -1, 1],[-1,1,-1],[1,-1,1]] :: Mat3 Double
+--   >>> 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
+  where
+    -- | Square and add up the entries of a row.
+    row_sum = element_sum . V.map (^2)
 
 
 -- Vector helpers. We want it to be easy to create low-dimension
--- column vectors.
-type Vec a b = Mat a D1 b
-
-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))
-
-dot :: (RealRing.C a,
-        Dim w ~ V.N1,
-        Vector v a,
-        Vector w a,
-        Vector w (v a),
-        Vector w (w a))
-       => Mat v w a
-       -> Mat v w a
+-- column vectors, which are nx1 matrices.
+
+-- | Convenient constructor for 2D vectors.
+--
+--   Examples:
+--
+--   >>> import Roots.Simple
+--   >>> let fst m = m !!! (0,0)
+--   >>> let snd m = m !!! (1,0)
+--   >>> let h = 0.5 :: Double
+--   >>> let g1 m = 1.0 + h NP.* exp(-((fst m)^2))/(1.0 + (snd m)^2)
+--   >>> let g2 m = 0.5 + h NP.* atan((fst m)^2 + (snd m)^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) -> Col1 a
+vec1d (x) = Mat (mk1 (mk1 x))
+
+vec2d :: (a,a) -> Col2 a
+vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
+
+vec3d :: (a,a,a) -> Col3 a
+vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
+
+vec4d :: (a,a,a,a) -> Col4 a
+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))
+
+-- 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) `mult` v2) !!! (0, 0)
+v1 `dot` v2 = unscalar $ ((transpose v1) * v2)
 
 
 -- | The angle between @v1@ and @v2@ in Euclidean space.
 --
 --   Examples:
 --
---   >>> let v1 = make2d (1.0, 0.0)
---   >>> let v2 = make2d (0.0, 1.0)
+--   >>> 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 ~ V.N1,
-          Vector w (w a),
-          Vector v [a],
-          Vector v a,
-          Vector w a,
-          Vector v (w a),
-          Vector w (v a),
+          m ~ S t,
+          Arity t,
           ToRational.C a)
-          => Mat v w a
-          -> Mat v w a
+          => Col m a
+          -> Col m a
           -> a
 angle v1 v2 =
   acos theta
   where
-   theta = (recip norms) * (v1 `dot` v2)
-   norms = (norm v1) * (norm v2)
+   theta = (recip norms) NP.* (v1 `dot` v2)
+   norms = (norm v1) NP.* (norm v2)
+
+
+-- | Retrieve the diagonal elements of the given matrix as a \"column
+--   vector,\" i.e. a m-by-1 matrix. We require the matrix to be
+--   square to avoid ambiguity in the return type which would ideally
+--   have dimension min(m,n) supposing an m-by-n matrix.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> diagonal m
+--   ((1),(5),(9))
+--
+diagonal :: (Arity m) => Mat m m a -> Col m a
+diagonal matrix =
+  construct lambda
+  where
+    lambda i _ = matrix !!! (i,i)
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the on-diagonal entries of @matrix@. The
+--   off-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> diagonal_part m
+--   ((1,0,0),(0,5,0),(0,0,9))
+--
+diagonal_part :: (Arity m, Ring.C a)
+         => Mat m m a
+         -> Mat m m a
+diagonal_part matrix =
+  construct lambda
+  where
+    lambda i j = if i == j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the on-diagonal and below-diagonal entries of
+--   @matrix@. The above-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> lt_part m
+--   ((1,0,0),(4,5,0),(7,8,9))
+--
+lt_part :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+lt_part matrix =
+  construct lambda
+  where
+    lambda i j = if i >= j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the below-diagonal entries of @matrix@. The on-
+--   and above-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> lt_part_strict m
+--   ((0,0,0),(4,0,0),(7,8,0))
+--
+lt_part_strict :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+lt_part_strict matrix =
+  construct lambda
+  where
+    lambda i j = if i > j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the on-diagonal and above-diagonal entries of
+--   @matrix@. The below-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> ut_part m
+--   ((1,2,3),(0,5,6),(0,0,9))
+--
+ut_part :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+ut_part = transpose . lt_part . transpose
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the above-diagonal entries of @matrix@. The on-
+--   and below-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> ut_part_strict m
+--   ((0,2,3),(0,0,6),(0,0,0))
+--
+ut_part_strict :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+ut_part_strict = transpose . lt_part_strict . transpose
+
+
+-- | Compute the trace of a square matrix, the sum of the elements
+--   which lie on its diagonal. We require the matrix to be
+--   square to avoid ambiguity in the return type which would ideally
+--   have dimension min(m,n) supposing an m-by-n matrix.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> trace m
+--   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
+
+
+-- | 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
+--   >>> zip2 m1 m2
+--   (((1,1)),((1,2)),((1,3)))
+--
+--   >>> 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 b -> Mat m n (a,b)
+zip2 m1 m2 =
+  construct lambda
+  where
+    lambda i j = (m1 !!! (i,j), m2 !!! (i,j))
+
+
+-- | 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
+--   >>> zipwith2 (^) c1 c2
+--   ((1),(32),(729))
+--
+zipwith2 :: Arity m
+           => (a -> a -> b)
+           -> Col m a
+           -> Col m a
+           -> Col m b
+zipwith2 f c1 c2 =
+  construct lambda
+  where
+    lambda i j = f (c1 !!! (i,j)) (c2 !!! (i,j))
+
+
+-- | Map a function over a matrix of any dimensions.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+--   >>> map2 (^2) m
+--   ((1,4),(9,16))
+--
+map2 :: (a -> b) -> Mat m n a -> Mat m n b
+map2 f (Mat rows) =
+  Mat $ V.map g rows
+  where
+    g = V.map f
+
+
+-- | Fold over the entire matrix passing the coordinates @i@ and @j@
+--   (of the row/column) to the accumulation function. The fold occurs
+--   from top-left to bottom-right.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
+--   18
+--
+ifoldl2 :: forall a b m n.
+           (Int -> Int -> b -> a -> b)
+        -> b
+        -> Mat m n a
+        -> b
+ifoldl2 f initial (Mat rows) =
+  V.ifoldl row_function initial rows
+  where
+    -- | The order that we need this in (so that @g idx@ makes sense)
+    --   is a little funny. So that we don't need to pass weird
+    --   functions into ifoldl2, we swap the second and third
+    --   arguments of @f@ calling the result @g@.
+    g :: Int -> b -> Int -> a -> b
+    g w x y = f w y x
+
+    row_function :: b -> Int -> Vec n a -> b
+    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@.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+--   >>> imap2 (\i j _ -> i+j) m
+--   ((0,1),(1,2))
+--
+imap2 :: (Int -> Int -> a -> b) -> Mat m n a -> Mat m n b
+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
+
+
+-- | 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
+