Fix the incorrect definition of "minor".

Add a "preminor" function that does what "minor" used to.
Implement cofactors and a matrix inverse in terms of them.

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 119d77089461dfae968320c79bd1375c31d46b41..2fe53401587f1fe25ec4ebc16919d1463c543691 100644 (file)
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -56,6 +56,7 @@ 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 )
@@ -73,20 +74,22 @@ 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 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 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 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
@@ -467,35 +470,50 @@ is_triangular :: (Ord a,
 is_triangular m = is_upper_triangular m || is_lower_triangular m
 
 
--- | Return the (i,j)th minor of 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
---   >>> minor m 0 0 :: Mat2 Int
+--   >>> preminor m 0 0 :: Mat2 Int
 --   ((5,6),(8,9))
---   >>> minor m 1 1 :: Mat2 Int
+--   >>> preminor m 1 1 :: Mat2 Int
 --   ((1,3),(7,9))
 --
-minor :: (m ~ S r,
-          n ~ S t,
-          Arity r,
-          Arity t)
-      => Mat m n a
+preminor :: (Arity m, Arity n)
+      => Mat (S m) (S n) a
       -> Int
       -> Int
-      -> Mat r t a
-minor (Mat rows) i j = m
+      -> 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 (Mat rows) = (V.head . V.head) rows
+  determinant = unscalar
 
 instance (Ord a,
           Ring.C a,
@@ -517,10 +535,8 @@ instance (Ord a,
         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)
+            sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (minor m 0 j)
               | j <- [0..(ncols m)-1] ]
 
 
@@ -1028,3 +1044,40 @@ set_idx matrix (i,j) newval =
       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
+