X-Git-Url: http://gitweb.michael.orlitzky.com/?p=numerical-analysis.git;a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=34920b4f025e7e2027588ab07c14d6772b679ab2;hp=63c0348cd9e6e0ce676ce5c17c1629890836a6e6;hb=4e464a486bef07db44de9c3d3fae0c8094401b09;hpb=3c5015c938c96f70b15c6292198a01390ee6540a

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 63c0348..34920b4 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -1,82 +1,202 @@
-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ExistentialQuantification #-}
 {-# 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 (
+  and,
   fromList,
+  head,
   length,
   map,
-  toList
-  )
-import Data.Vector.Fixed.Internal (arity)
+  maximum,
+  replicate,
+  toList,
+  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
 
-import Linear.Vector
+instance (Eq a) => Eq (Mat m n a) where
+  -- | Compare a row at a time.
+  --
+  --   Examples:
+  --
+  --   >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> let m2 = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> let m3 = fromList [[5,6],[7,8]] :: Mat2 Int
+  --   >>> m1 == m2
+  --   True
+  --   >>> m1 == m3
+  --   False
+  --
+  (Mat rows1) == (Mat rows2) =
+    V.and $ V.zipWith comp rows1 rows2
+    where
+      -- Compare a row, one column at a time.
+      comp row1 row2 = V.and (V.zipWith (==) row1 row2)
+
+
+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
+  --   who insists his vector lengths be statically checked at
+  --   compile-time).
+  --
+  --   Examples:
+  --
+  --   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> show m
+  --   ((1,2),(3,4))
+  --
+  show (Mat rows) =
+    "(" ++ (intercalate "," (V.toList row_strings)) ++ ")"
+    where
+      row_strings = V.map show_vector rows
+      show_vector v1 =
+        "(" ++ (intercalate "," element_strings) ++ ")"
+        where
+          v1l = V.toList v1
+          element_strings = P.map show v1l
 
-type Mat v w a = Vn v (Vn w a)
-type Mat2 a = Mat Vec2D Vec2D a
-type Mat3 a = Mat Vec3D Vec3D a
-type Mat4 a = Mat Vec4D Vec4D a
 
 -- | Convert a matrix to a nested list.
-toList :: (Vector v (Vn w a), Vector w a) => Mat v w a -> [[a]]
-toList m = map V.toList (V.toList m)
+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 (Vn w a), Vector w a) => [[a]] -> Mat v w a
-fromList vs = V.fromList $ map V.fromList vs
+fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
+fromList vs = Mat (V.fromList $ map V.fromList vs)
 
 
 -- | Unsafe indexing.
-(!!!) :: (Vector v (Vn w a), 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.
-(!!?) :: (Vector v (Vn w a), Vector w a) => Mat v w a
-                                         -> (Int, Int)
-                                         -> Maybe a
-(!!?) m (i, j)
+(!!?) :: Mat m n a -> (Int, Int) -> Maybe a
+(!!?) m@(Mat rows) (i, j)
   | i < 0 || j < 0 = Nothing
-  | i > V.length m = Nothing
+  | i > V.length rows = Nothing
   | otherwise = if j > V.length (row m j)
                 then Nothing
                 else Just $ (row m j) ! j
 
 
 -- | The number of rows in the matrix.
-nrows :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-nrows = V.length
+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 v (Vn 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 :: (Vector v (Vn w a), Vector w a) => Mat v w a
-                                       -> Int
-                                       -> Vn w a
-row m i = m ! i
+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, Vector v (Vn w a), Vector w a) => Mat v w a
-                                                      -> Int
-                                                      -> Vn v a
-column m j =
-  V.map (element j) m
+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
+    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.
@@ -89,16 +209,12 @@ column m j =
 --   >>> transpose m
 --   ((1,3),(2,4))
 --
-transpose :: (Vector v (Vn w a),
-              Vector w (Vn v a),
-              Vector v a,
-              Vector w a)
-             => Mat v w a
-             -> Mat w v a
-transpose m = V.fromList column_list
+transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
+transpose m = Mat $ V.fromList column_list
   where
     column_list = [ column m i | i <- [0..(ncols m)-1] ]
 
+
 -- | Is @m@ symmetric?
 --
 --   Examples:
@@ -111,13 +227,7 @@ transpose m = V.fromList column_list
 --   >>> symmetric m2
 --   False
 --
-symmetric :: (Vector v (Vn 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)
 
@@ -127,26 +237,32 @@ 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 (Vn w a),
-              Vector w a)
-             => (Int -> Int -> a)
-             -> Mat v w a
-construct lambda = 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
@@ -157,43 +273,497 @@ construct lambda = rows
 --   >>> 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)
+--   >>> (transpose (cholesky m1)) * (cholesky m1)
 --   ((20.000000000000004,-1.0),(-1.0,20.0))
 --
-cholesky :: forall a v w.
-            (RealFloat a,
-             Vector v (Vn w a),
-             Vector w 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
-    r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)**2 | k <- [0..i-1]])
+    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
+--
+--   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 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 Vec2D Vec3D Int
---   >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat Vec3D Vec2D Int
---   >>> m1 `mult` m2
+--   >>> 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 :: (Num a,
-         Vector v (Vn w a),
-         Vector w a,
-         Vector w (Vn z a),
-         Vector z a,
-         Vector v (Vn 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, Arity m, Arity n) => Additive.C (Mat m n a) where
+
+  (Mat rows1) + (Mat rows2) =
+    Mat $ V.zipWith (V.zipWith (+)) rows1 rows2
+
+  (Mat rows1) - (Mat rows2) =
+    Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
+
+  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
+  -- The first * is ring multiplication, the second is matrix
+  -- multiplication.
+  m1 * m2 = m1 * m2
+
+
+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
+
+
+instance (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
+  --   matrices.
+  --
+  --   Examples:
+  --
+  --   >>> let v1 = vec2d (3,4)
+  --   >>> norm_p 1 v1
+  --   7.0
+  --   >>> norm_p 2 v1
+  --   5.0
+  --
+  norm_p p (Mat rows) =
+    (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.
+  --
+  --   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, 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))
+
+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)
+
+
+-- | The angle between @v1@ and @v2@ in Euclidean space.
+--
+--   Examples:
+--
+--   >>> 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,
+          n ~ N1,
+          m ~ S t,
+          Arity t,
+          ToRational.C 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