From 1f64a1a33b2636ef2e863a0b577c8d8d50233580 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 24 Feb 2013 02:41:03 -0500
Subject: [PATCH] Replace the whole Matrix implementation with something a
 little better. Add the Linear.System module. Get all doctests working again.
 Update .ghci/cabal files.

---
 .ghci                    |   4 +-
 numerical-analysis.cabal |   4 +-
 src/Linear/Matrix.hs     | 294 +++++++++++++++++----------------------
 src/Linear/System.hs     |  37 ++---
 src/Linear/Vector.hs     |  49 ++-----
 5 files changed, 153 insertions(+), 235 deletions(-)

diff --git a/.ghci b/.ghci
index 16cb969..9a6a4d4 100644
--- a/.ghci
+++ b/.ghci
@@ -7,6 +7,7 @@
   src/Integration/Simpson.hs
   src/Integration/Trapezoid.hs
   src/Linear/Matrix.hs
+  src/Linear/System.hs
   src/Linear/Vector.hs
   src/Misc.hs
   src/Normed.hs
@@ -17,8 +18,9 @@
 import BigFloat
 import Integration.Simpson
 import Integration.Trapezoid
-import Linear.Vector
 import Linear.Matrix
+import Linear.System
+import Linear.Vector
 import Misc
 import Normed
 import ODE.IVP
diff --git a/numerical-analysis.cabal b/numerical-analysis.cabal
index 0024786..e459791 100644
--- a/numerical-analysis.cabal
+++ b/numerical-analysis.cabal
@@ -18,8 +18,8 @@ data-files: makefile
 
 library
   exposed-modules: Integration.Simpson, Integration.Trapezoid,
-    Linear.Matrix, Linear.Vector, Misc, Normed, ODE.IVP, Roots.Simple,
-    Roots.Fast
+    Linear.Matrix, Linear.System, Linear.Vector, Misc, Normed, ODE.IVP,
+    Roots.Simple, Roots.Fast
 
   build-depends:
     base              >= 3 && < 5,
diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 5bb5519..7452007 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -6,26 +6,42 @@
 {-# 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,
   N1,
-  Vector
+  N2,
+  N3,
+  N4,
+  N5,
+  S,
+  Z,
+  mk1,
+  mk2,
+  mk3,
+  mk4,
+  mk5
   )
 import qualified Data.Vector.Fixed as V (
   and,
   fromList,
   length,
   map,
+  maximum,
   replicate,
   toList,
   zipWith
   )
-import Data.Vector.Fixed.Internal (Arity, arity, S)
+import Data.Vector.Fixed.Boxed (Vec)
+import Data.Vector.Fixed.Internal (Arity, arity)
 import Linear.Vector
 import Normed
 
@@ -41,16 +57,14 @@ 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
+data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
+type Mat1 a = Mat N1 N1 a
+type Mat2 a = Mat N2 N2 a
+type Mat3 a = Mat N3 N3 a
+type Mat4 a = Mat N4 N4 a
+type Mat5 a = Mat N5 N5 a
 
--- 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
+instance (Eq a) => Eq (Mat m n a) where
   -- | Compare a row at a time.
   --
   --   Examples:
@@ -70,7 +84,7 @@ 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 +108,21 @@ 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
@@ -119,27 +132,30 @@ 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 @j@th column of @m@. Unsafe.
-column :: (Vector v a) => Mat v w a -> Int -> v a
+column :: Mat m n a -> Int -> (Vec m a)
 column (Mat rows) j =
   V.map (element j) rows
   where
     element = flip (!)
 
 
+
+
 -- | 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.
@@ -152,11 +168,7 @@ 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 :: (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] ]
@@ -174,13 +186,7 @@ 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)
 
@@ -198,19 +204,17 @@ symmetric m =
 --   >>> 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 :: forall m n a. (Arity m, Arity n)
+          => (Int -> Int -> a) -> Mat m n a
 construct lambda = Mat rows
   where
     -- The arity trick is used in Data.Vector.Fixed.length.
-    imax = (arity (undefined :: Dim v)) - 1
-    jmax = (arity (undefined :: Dim w)) - 1
+    imax = (arity (undefined :: m)) - 1
+    jmax = (arity (undefined :: n)) - 1
     row' i = V.fromList [ lambda i j | j <- [0..jmax] ]
     rows = V.fromList [ row' i | i <- [0..imax] ]
 
+
 -- | Given a positive-definite matrix @m@, computes the
 --   upper-triangular matrix @r@ with (transpose r)*r == m and all
 --   values on the diagonal of @r@ positive.
@@ -223,13 +227,8 @@ 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
@@ -252,7 +251,8 @@ cholesky m = construct r
 --   >>> is_upper_triangular m
 --   True
 --
-is_upper_triangular :: (Eq a, Ring.C a, Vector w a) => Mat v w a -> Bool
+is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n)
+                    => Mat m n a -> Bool
 is_upper_triangular m =
   and $ concat results
   where
@@ -279,10 +279,9 @@ is_upper_triangular m =
 --
 is_lower_triangular :: (Eq a,
                         Ring.C a,
-                        Vector w a,
-                        Vector w (v a),
-                        Vector v a)
-                    => Mat v w a
+                        Arity m,
+                        Arity n)
+                    => Mat m n a
                     -> Bool
 is_lower_triangular = is_upper_triangular . transpose
 
@@ -306,10 +305,9 @@ is_lower_triangular = is_upper_triangular . transpose
 --
 is_triangular :: (Eq a,
                   Ring.C a,
-                  Vector w a,
-                  Vector w (v a),
-                  Vector v a)
-               => Mat v w a
+                  Arity m,
+                  Arity n)
+              => Mat m n a
               -> Bool
 is_triangular m = is_upper_triangular m || is_lower_triangular m
 
@@ -324,73 +322,60 @@ is_triangular m = is_upper_triangular m || is_lower_triangular m
 --   >>> minor m 1 1 :: Mat2 Int
 --   ((1,3),(7,9))
 --
-minor :: (Dim v ~ S (Dim u),
-          Dim w ~ S (Dim z),
-          Vector z a,
-          Vector u (w a),
-          Vector u (z a))
-      => Mat v w a
+minor :: (m ~ S r,
+          n ~ S t,
+          Arity r,
+          Arity t)
+      => Mat m n a
       -> Int
       -> Int
-      -> Mat u z a
+      -> Mat r t a
 minor (Mat rows) i j = m
   where
     rows' = delete rows i
     m = Mat $ V.map ((flip delete) j) rows'
 
 
-determinant :: (Eq a,
-                Ring.C a,
-                Vector w a,
-                Vector w (v a),
-                Vector v a,
-                Dim v ~ S r,
-                Dim w ~ S t)
-             => Mat v w a
-             -> a
-determinant m
-  | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
-  | otherwise = undefined --determinant_recursive m
-
-{-
-determinant_recursive :: forall v w a r c.
-                         (Eq a,
-                          Ring.C a,
-                          Vector w a)
-                       => Mat (v r) (w c) a
-                       -> a
-determinant_recursive m
-  | (ncols m) == 0 || (nrows m) == 0 = error "don't do that"
-  | (ncols m) == 1 && (nrows m) == 1 = m !!! (0,0) -- Base case
-  | otherwise =
-      sum [ (-1)^(1+(toInteger j)) NP.* (m' 1 j) NP.* (det_minor 1 j)
-            | j <- [0..(ncols m)-1] ]
-      where
-        m' i j = m !!! (i,j)
-
-        det_minor :: Int -> Int -> a
-        det_minor i j = determinant (minor m 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 m = m !!! (0,0)
+
+instance (Eq a, Ring.C a, Arity m) => Determined (Mat m m) a where
+  determinant _ = undefined
+
+instance (Eq a, Ring.C a, Arity n)
+         => Determined (Mat (S (S n)) (S (S n))) a where
+  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)^(1+(toInteger j)) NP.* (m' 0 j) NP.* (det_minor 0 j)
+              | j <- [0..(ncols m)-1] ]
+
+
 
 -- | Matrix multiplication. Our 'Num' instance doesn't define one, and
 --   we need additional restrictions on the result type anyway.
 --
 --   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 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 =
@@ -398,10 +383,7 @@ 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
@@ -412,20 +394,13 @@ 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
@@ -433,11 +408,9 @@ 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
+          Arity m,
+          Arity n)
+         => Normed (Mat (S m) (S n) a) where
   -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
   --   all matrices as big vectors.
   --
@@ -455,9 +428,7 @@ instance (Algebraic.C a,
       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:
   --
@@ -465,11 +436,8 @@ 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
 
 
 
@@ -483,41 +451,40 @@ 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))
 --
-vec1d :: (a) -> Mat D1 D1 a
-vec1d (x) = Mat (D1 (D1 x))
+vec1d :: (a) -> Mat N1 N1 a
+vec1d (x) = Mat (mk1 (mk1 x))
 
-vec2d :: (a,a) -> Mat D2 D1 a
-vec2d (x,y) = Mat (D2 (D1 x) (D1 y))
+vec2d :: (a,a) -> Mat N2 N1 a
+vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
 
-vec3d :: (a,a,a) -> Mat D3 D1 a
-vec3d (x,y,z) = Mat (D3 (D1 x) (D1 y) (D1 z))
+vec3d :: (a,a,a) -> Mat N3 N1 a
+vec3d (x,y,z) = Mat (mk3 (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))
+vec4d :: (a,a,a,a) -> Mat N4 N1 a
+vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
+
+vec5d :: (a,a,a,a,a) -> Mat N5 N1 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 ~ 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 -> Mat N1 N1 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)
 
@@ -533,17 +500,12 @@ v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
 --
 angle :: (Transcendental.C a,
           RealRing.C a,
-          Dim w ~ 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
diff --git a/src/Linear/System.hs b/src/Linear/System.hs
index e0fdf1c..58d8d78 100644
--- a/src/Linear/System.hs
+++ b/src/Linear/System.hs
@@ -5,7 +5,7 @@
 module Linear.System
 where
 
-import Data.Vector.Fixed (Dim, N1, Vector)
+import Data.Vector.Fixed (Arity, N1)
 
 import Linear.Matrix
 
@@ -13,7 +13,6 @@ import NumericPrelude hiding ((*), abs)
 import qualified NumericPrelude as NP ((*))
 import qualified Algebra.Field as Field
 
-import Debug.Trace (trace, traceShow)
 
 -- | Solve the system m' * x = b', where m' is upper-triangular. Will
 --   probably crash if m' is non-singular. The result is the vector x.
@@ -21,27 +20,21 @@ import Debug.Trace (trace, traceShow)
 --   Examples:
 --
 --   >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double
---   >>> let b = vec3d (1,2,3)
+--   >>> let b = vec3d (1, 2, 3::Double)
 --   >>> forward_substitute identity b
 --   ((1.0),(2.0),(3.0))
 --   >>> (forward_substitute identity b) == b
 --   True
 --
 --   >>> let m = fromList [[1,0],[1,1]] :: Mat2 Double
---   >>> let b = vec2d (1,1)
+--   >>> let b = vec2d (1, 1::Double)
 --   >>> forward_substitute m b
 --   ((1.0),(0.0))
 --
-forward_substitute :: forall a v w z.
-                      (Show a, Field.C a,
-                       Vector z a,
-                       Vector w (z a),
-                       Vector w a,
-                       Dim z ~ N1,
-                       v ~ w)
-                   => Mat v w a
-                   -> Mat w z a
-                   -> Mat w z a
+forward_substitute :: forall a m. (Field.C a, Arity m)
+                   => Mat m m a
+                   -> Mat m N1 a
+                   -> Mat m N1 a
 forward_substitute m' b' = x'
   where
     x' = construct lambda
@@ -72,22 +65,16 @@ forward_substitute m' b' = x'
 --   Examples:
 --
 --   >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double
---   >>> let b = vec3d (1,2,3)
+--   >>> let b = vec3d (1, 2, 3::Double)
 --   >>> backward_substitute identity b
 --   ((1.0),(2.0),(3.0))
 --   >>> (backward_substitute identity b) == b
 --   True
 --
-backward_substitute :: (Show a, Field.C a,
-                        Vector z a,
-                        Vector v (w a),
-                        Vector w (z a),
-                        Vector w a,
-                        Dim z ~ N1,
-                        v ~ w)
-                    => Mat v w a
-                    -> Mat w z a
-                    -> Mat w z a
+backward_substitute :: (Field.C a, Arity m)
+                    => Mat m m a
+                    -> Mat m N1 a
+                    -> Mat m N1 a
 backward_substitute m b =
   forward_substitute (transpose m) b
 
diff --git a/src/Linear/Vector.hs b/src/Linear/Vector.hs
index ec6ebcb..aa1568b 100644
--- a/src/Linear/Vector.hs
+++ b/src/Linear/Vector.hs
@@ -9,17 +9,12 @@ where
 
 import Data.Vector.Fixed (
   Dim,
-  Fun(..),
   N1,
-  N2,
-  N3,
   N4,
   N5,
   S,
   Vector(..),
-  construct,
   fromList,
-  inspect,
   toList,
   )
 import qualified Data.Vector.Fixed as V (
@@ -31,43 +26,13 @@ type Vec1 = Vec N1
 type Vec4 = Vec N4
 type Vec5 = Vec N5
 
--- * Low-dimension vector wrappers.
---
--- These wrappers are instances of 'Vector', so they inherit all of
--- the userful instances defined above. But, they use fixed
--- constructors, so you can pattern match out the individual
--- components.
-
-data D1 a = D1 a deriving (Show, Eq)
-type instance Dim D1 = N1
-instance Vector D1 a where
-  inspect (D1 x) (Fun f) = f x
-  construct = Fun D1
-
-data D2 a = D2 a a deriving (Show, Eq)
-type instance Dim D2 = N2
-instance Vector D2 a where
-  inspect (D2 x y) (Fun f) = f x y
-  construct = Fun D2
-
-data D3 a = D3 a a a deriving (Show, Eq)
-type instance Dim D3 = N3
-instance Vector D3 a where
-  inspect (D3 x y z) (Fun f) = f x y z
-  construct = Fun D3
-
-data D4 a = D4 a a a a deriving (Show, Eq)
-type instance Dim D4 = N4
-instance Vector D4 a where
-  inspect (D4 w x y z) (Fun f) = f w x y z
-  construct = Fun D4
-
 
 -- | Unsafe indexing.
 --
 --   Examples:
 --
---   >>> let v1 = D2 1 2
+--   >>> import Data.Vector.Fixed (mk2)
+--   >>> let v1 = mk2 1 2 :: Vec2 Int
 --   >>> v1 ! 1
 --   2
 --
@@ -78,7 +43,8 @@ instance Vector D4 a where
 --
 --   Examples:
 --
---   >>> let v1 = D3 1 2 3
+--   >>> import Data.Vector.Fixed (mk3)
+--   >>> let v1 = mk3 1 2 3 :: Vec3 Int
 --   >>> v1 !? 2
 --   Just 3
 --   >>> v1 !? 3
@@ -94,9 +60,10 @@ instance Vector D4 a where
 --
 --   Examples:
 --
---   >>> let b = D3 1 2 3
---   >>> delete b 1 :: D2 Int
---   D2 1 3
+--   >>> import Data.Vector.Fixed (mk3)
+--   >>> let b = mk3 1 2 3 :: Vec3 Int
+--   >>> delete b 1 :: Vec2 Int
+--   fromList [1,3]
 --
 delete :: (Vector v a, Vector w a, Dim v ~ S (Dim w)) => v a -> Int -> w a
 delete v1 idx =
-- 
2.44.2