X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FNormed.hs;h=6f34a8dca406c13393896179b6b08c023874e9a3;hb=be2ec3ca8e6fda229e3ca608dcc75e085b3a0b0f;hp=8554bd97f2eba0bd66a4b458a05cdb334f2e4f87;hpb=fe73028041fe3becce6ce1ff268181d55d54a011;p=numerical-analysis.git

diff --git a/src/Normed.hs b/src/Normed.hs
index 8554bd9..6f34a8d 100644
--- a/src/Normed.hs
+++ b/src/Normed.hs
@@ -1,5 +1,6 @@
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE RebindableSyntax #-}
+
 -- | The 'Normed' class represents elements of a normed vector
 --   space. We define instances for all common numeric types.
 module Normed
@@ -7,28 +8,29 @@ where
 
 import BigFloat
 
-import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
-import Algebra.ToInteger
-
--- Since the norm is defined on a vector space, we should be able to
--- add and subtract anything on which a norm is defined. Of course
--- 'Num' is a bad choice here, but we really prefer to use the normal
--- addition and subtraction operators.
-class (Algebra.Ring.C a, Algebra.Absolute.C a) => Normed a where
-  norm_p :: (Algebra.ToInteger.C c,
-             Algebra.Field.C b,
-             Algebra.Absolute.C b)
-            => c -> a -> b
-
-  norm_infty :: (Algebra.Field.C b,
-                 Algebra.Absolute.C b)
-                => a -> b
+import NumericPrelude hiding ( abs )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Absolute as Absolute
+import qualified Algebra.Algebraic as Algebraic
+import Algebra.Algebraic ( root )
+import qualified Algebra.RealField as RealField
+import qualified Algebra.ToInteger as ToInteger
+import qualified Algebra.ToRational as ToRational ( C )
+import Data.Vector.Fixed ( S, Z )
+import qualified Data.Vector.Fixed as V (
+  Arity,
+  map,
+  maximum )
+import Data.Vector.Fixed.Boxed ( Vec )
+
+import Linear.Vector ( element_sum )
+
+class Normed a where
+  norm_p :: (ToInteger.C c, Algebraic.C b, Absolute.C b) => c -> a -> b
+  norm_infty :: (RealField.C b) => a -> b
 
   -- | The "usual" norm. Defaults to the Euclidean norm.
-  norm :: (Algebra.Field.C b, Algebra.Absolute.C b) => a -> b
+  norm :: (Algebraic.C b, Absolute.C b) => a -> b
   norm = norm_p (2 :: Integer)
 
 -- Define instances for common numeric types.
@@ -51,3 +53,38 @@ instance Normed Float where
 instance Normed Double where
   norm_p _ = abs . fromRational' . toRational
   norm_infty = abs . fromRational' . toRational
+
+
+-- | 'Normed' instance for vectors of length zero. These are easy.
+instance Normed (Vec Z a) where
+  norm_p _ = const (fromInteger 0)
+  norm_infty = const (fromInteger 0)
+
+
+-- | 'Normed' instance for vectors of length greater than zero. We
+--   need to know that the length is non-zero in order to invoke
+--   V.maximum. We will generally be working with n-by-1 /matrices/
+--   instead of vectors, but sometimes it's convenient to have these
+--   instances anyway.
+--
+--   Examples:
+--
+--   >>> import Data.Vector.Fixed (mk3)
+--   >>> import Linear.Vector (Vec3)
+--   >>> let b = mk3 1 2 3 :: Vec3 Double
+--   >>> norm_p 1 b :: Double
+--   6.0
+--   >>> norm b == sqrt 14
+--   True
+--   >>> norm_infty b :: Double
+--   3.0
+--
+instance (V.Arity n, Absolute.C a, ToRational.C a, Ord a)
+         => Normed (Vec (S n) a) where
+  norm_p p x =
+    (root p') $ element_sum $ V.map element_function x
+    where
+      element_function y = fromRational' $ (toRational y)^p'
+      p' = toInteger p
+
+  norm_infty x = fromRational' $ toRational $ V.maximum $ V.map abs x