X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=ea6bc5772422c620daa3057c0177a946442fc690;hb=ae914d13235a4582077a5cb2b1edd630d9c6ad62;hp=66b22f9f9d964e843e596678c0f7ebe34e8e9a22;hpb=af25e6f32f6787a747be63093adc10915ad60068;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 66b22f9..ea6bc57 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -35,7 +35,6 @@ import Data.Vector.Fixed (
   )
 import qualified Data.Vector.Fixed as V (
   and,
-  foldl,
   fromList,
   head,
   length,
@@ -45,22 +44,23 @@ import qualified Data.Vector.Fixed as V (
   toList,
   zipWith
   )
-import Data.Vector.Fixed.Boxed (Vec)
-import Data.Vector.Fixed.Cont (Arity, arity)
-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
+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 )
 
 data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
 type Mat1 a = Mat N1 N1 a
@@ -252,21 +252,26 @@ cholesky m = construct r
 
 
 -- | Returns True if the given matrix is upper-triangular, and False
---   otherwise.
+--   otherwise. The parameter @epsilon@ lets the caller choose a
+--   tolerance.
 --
 --   Examples:
 --
---   >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+--   >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
 --   >>> is_upper_triangular m
 --   False
---
---   >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
---   >>> is_upper_triangular m
+--   >>> is_upper_triangular' 1e-10 m
 --   True
 --
-is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n)
-                    => Mat m n a -> Bool
-is_upper_triangular m =
+--   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] ]
@@ -274,11 +279,36 @@ is_upper_triangular m =
     test :: Int -> Int -> Bool
     test i j
       | i <= j = True
-      | otherwise = m !!! (i,j) == 0
+      -- 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.
+--   otherwise. This is a specialized version of 'is_lower_triangular\''
+--   with @epsilon = 0@.
 --
 --   Examples:
 --
@@ -290,8 +320,9 @@ is_upper_triangular m =
 --   >>> is_lower_triangular m
 --   False
 --
-is_lower_triangular :: (Eq a,
+is_lower_triangular :: (Ord a,
                         Ring.C a,
+                        Absolute.C a,
                         Arity m,
                         Arity n)
                     => Mat m n a
@@ -299,6 +330,29 @@ is_lower_triangular :: (Eq a,
 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.
 --
@@ -316,8 +370,9 @@ is_lower_triangular = is_upper_triangular . transpose
 --   >>> is_triangular m
 --   False
 --
-is_triangular :: (Eq a,
+is_triangular :: (Ord a,
                   Ring.C a,
+                  Absolute.C a,
                   Arity m,
                   Arity n)
               => Mat m n a
@@ -355,8 +410,9 @@ class (Eq a, Ring.C a) => Determined p a where
 instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
   determinant (Mat rows) = (V.head . V.head) rows
 
-instance (Eq a,
+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
@@ -430,8 +486,8 @@ instance (Algebraic.C a,
           ToRational.C a,
           Arity m)
          => Normed (Mat (S m) N1 a) where
-  -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
-  --   all matrices as big vectors.
+  -- | Generic p-norms for vectors in R^n that are represented as nx1
+  --   matrices.
   --
   --   Examples:
   --
@@ -475,15 +531,10 @@ instance (Algebraic.C a,
 --
 frobenius_norm :: (Algebraic.C a, Ring.C a) => Mat m n a -> a
 frobenius_norm (Mat rows) =
-  sqrt $ vsum $ V.map row_sum rows
+  sqrt $ element_sum $ V.map row_sum rows
   where
-    -- | The \"sum\" function defined in fixed-vector requires a 'Num'
-    --   constraint whereas we want to use the classes from
-    --   numeric-prelude.
-    vsum = V.foldl (+) (fromInteger 0)
-
     -- | Square and add up the entries of a row.
-    row_sum = vsum . V.map (^2)
+    row_sum = element_sum . V.map (^2)
 
 
 -- Vector helpers. We want it to be easy to create low-dimension