X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=6d13a693b71e37631e8600f87652f6c9eb73b883;hb=a96be0c7818e0408b4aac9076df131c5b3ec6ff4;hp=4f2e845da6b97380b703441c1fc6963c568cb943;hpb=2f54e89d36e835c58efcc281741632d457859b20;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 4f2e845..6d13a69 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -2,55 +2,102 @@
 {-# 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
+  (!),
+  N1,
+  N2,
+  N3,
+  N4,
+  N5,
+  S,
+  Z,
+  generate,
+  mk1,
+  mk2,
+  mk3,
+  mk4,
+  mk5
   )
 import qualified Data.Vector.Fixed as V (
-  N1,
   and,
   fromList,
+  head,
+  ifoldl,
   length,
   map,
+  maximum,
   replicate,
   toList,
-  zipWith
-  )
-import Data.Vector.Fixed.Internal (Arity, arity, S)
-import Linear.Vector
-import Normed
-
-import NumericPrelude hiding ((*), abs)
-import qualified NumericPrelude as NP ((*))
-import qualified Algebra.Algebraic as Algebraic
-import Algebra.Algebraic (root)
-import qualified Algebra.Additive as Additive
-import qualified Algebra.Ring as Ring
-import qualified Algebra.Module as Module
-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 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.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 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 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
+
+
+instance (Eq a) => Eq (Mat m n a) where
   -- | Compare a row at a time.
   --
   --   Examples:
@@ -70,7 +117,7 @@ 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
@@ -94,22 +141,21 @@ 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
+(!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
 (!!!) m (i, j) = (row m i) ! j
 
 -- | Safe indexing.
-(!!?) :: Mat v w a -> (Int, Int) -> Maybe a
+(!!?) :: Mat m n a -> (Int, Int) -> Maybe a
 (!!?) m@(Mat rows) (i, j)
   | i < 0 || j < 0 = Nothing
   | i > V.length rows = Nothing
@@ -119,30 +165,46 @@ fromList vs = Mat (V.fromList $ map V.fromList vs)
 
 
 -- | 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 m n a -> Int -> (Vec n 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
+--column :: Mat m n a -> Int -> (Vec m a)
+--column (Mat rows) j =
+--  V.map (element j) rows
+--  where
+--    element = flip (!)
+
+
+-- | 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.
 --
@@ -152,14 +214,11 @@ 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?
@@ -174,13 +233,7 @@ 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)
 
@@ -190,26 +243,32 @@ 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
@@ -223,13 +282,8 @@ construct lambda = Mat rows
 --   >>> (transpose (cholesky m1)) * (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)
+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
@@ -239,25 +293,207 @@ cholesky m = construct r
           | 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
+--
+--   TODO:
+--
+--     1. Don't cheat with lists.
+--
+is_upper_triangular' :: (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
+  where
+    results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
+
+    test :: Int -> Int -> Bool
+    test i j
+      | i <= j = True
+      -- use "less than or equal to" so zero is a valid epsilon
+      | otherwise = abs (m !!! (i,j)) <= epsilon
+
+
+-- | Returns True if the given matrix is upper-triangular, and False
+--   otherwise. A specialized version of 'is_upper_triangular\'' with
+--   @epsilon = 0@.
+--
+--   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
+--
+--   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)
+                    => Mat m n a -> Bool
+is_upper_triangular = is_upper_triangular' 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@.
+--
+--   Examples:
+--
+--   >>> 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 :: (Ord a,
+                        Ring.C a,
+                        Absolute.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
+
+
+-- | Return the (i,j)th minor of m.
 --
 --   Examples:
 --
---   >>> let m1 = fromList [[1,2,3], [4,5,6]]  :: Mat D2 D3 Int
---   >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat D3 D2 Int
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> minor m 0 0 :: Mat2 Int
+--   ((5,6),(8,9))
+--   >>> minor m 1 1 :: Mat2 Int
+--   ((1,3),(7,9))
+--
+minor :: (m ~ S r,
+          n ~ S t,
+          Arity r,
+          Arity t)
+      => Mat m n a
+      -> Int
+      -> Int
+      -> Mat r t a
+minor (Mat rows) i j = m
+  where
+    rows' = delete rows i
+    m = Mat $ V.map ((flip delete) j) rows'
+
+
+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
+
+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)
+
+          det_minor i j = determinant (minor m i j)
+
+          determinant_recursive =
+            sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 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))
 --
 infixl 7 *
-(*) :: (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
+(*) :: (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 =
@@ -265,10 +501,7 @@ infixl 7 *
 
 
 
-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
@@ -279,20 +512,13 @@ 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
+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 (Ring.C a,
-          Vector v (w a),
-          Vector w a)
-         => Module.C a (Mat v w a) where
+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
@@ -300,13 +526,10 @@ instance (Ring.C a,
 
 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
-  -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
-  --   all matrices as big vectors.
+          Arity m)
+         => Normed (Mat (S m) N1 a) where
+  -- | Generic p-norms for vectors in R^n that are represented as nx1
+  --   matrices.
   --
   --   Examples:
   --
@@ -317,14 +540,12 @@ instance (Algebraic.C a,
   --   5.0
   --
   norm_p p (Mat rows) =
-    (root p') $ sum [(fromRational' $ toRational x)^p' | x <- xs]
+    (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
     where
       p' = toInteger p
       xs = concat $ V.toList $ V.map V.toList rows
 
-  -- | The infinity norm. We don't use V.maximum here because it
-  --   relies on a type constraint that the vector be non-empty and I
-  --   don't know how to pattern match it away.
+  -- | The infinity norm.
   --
   --   Examples:
   --
@@ -332,14 +553,30 @@ instance (Algebraic.C a,
   --   >>> norm_infty v1
   --   5
   --
-  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
-
+  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
@@ -350,38 +587,40 @@ instance (Algebraic.C a,
 --   Examples:
 --
 --   >>> import Roots.Simple
+--   >>> let fst m = m !!! (0,0)
+--   >>> let snd m = m !!! (1,0)
 --   >>> let h = 0.5 :: Double
---   >>> let g1 (Mat (D2 (D1 x) (D1 y))) = 1.0 + h NP.* exp(-(x^2))/(1.0 + y^2)
---   >>> let g2 (Mat (D2 (D1 x) (D1 y))) = 0.5 + h NP.* atan(x^2 + y^2)
+--   >>> 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))
 --
-vec2d :: (a,a) -> Mat D2 D1 a
-vec2d (x,y) = Mat (D2 (D1 x) (D1 y))
+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))
 
-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) -> Col4 a
+vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 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))
+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 -> Mat D1 D1 a
-scalar x = Mat (D1 (D1 x))
-
-dot :: (RealRing.C a,
-        Dim w ~ V.N1,
-        Dim v ~ S n,
-        Vector v a,
-        Vector w a,
-        Vector w (v a),
-        Vector w (w a))
-       => Mat v w a
-       -> Mat v w a
+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
        -> a
 v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
 
@@ -397,20 +636,221 @@ v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
 --
 angle :: (Transcendental.C a,
           RealRing.C a,
-          Dim w ~ V.N1,
-          Dim v ~ S n,
-          Vector w (w a),
-          Vector v [a],
-          Vector v a,
-          Vector w a,
-          Vector v (w a),
-          Vector w (v a),
+          n ~ N1,
+          m ~ S t,
+          Arity t,
           ToRational.C a)
-          => Mat v w a
-          -> Mat v w a
+          => Mat m n a
+          -> Mat m n a
           -> a
 angle v1 v2 =
   acos theta
   where
    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 column matrices.
+--
+--   Examples:
+--
+--   >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
+--   >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
+--   >>> colzip m1 m2
+--   (((1,1)),((1,2)),((1,3)))
+--
+colzip :: Arity m => Col m a -> Col m a -> Col m (a,a)
+colzip c1 c2 =
+  construct lambda
+  where
+    lambda i j = (c1 !!! (i,j), c2 !!! (i,j))
+
+
+-- | Zip together two column 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
+--   ((1),(32),(729))
+--
+colzipwith :: Arity m
+           => (a -> a -> b)
+           -> Col m a
+           -> Col m a
+           -> Col m b
+colzipwith 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.
+--
+--   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
+
+