X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FNormed.hs;h=21c275fc8ecdecadcc849c65b45f6fe02e0c3d0d;hb=ae914d13235a4582077a5cb2b1edd630d9c6ad62;hp=b60c2b12fe84d51e0bf83b60ce0f958af55f7c10;hpb=74e79199dcfec0639133ae9990dc33a2c5a095f0;p=numerical-analysis.git

diff --git a/src/Normed.hs b/src/Normed.hs
index b60c2b1..21c275f 100644
--- a/src/Normed.hs
+++ b/src/Normed.hs
@@ -1,29 +1,94 @@
 {-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE RebindableSyntax #-}
 
 -- | The 'Normed' class represents elements of a normed vector
 --   space. We define instances for all common numeric types.
 module Normed
 where
 
-import Data.Number.BigFloat
+import BigFloat
 
+import NumericPrelude hiding ( abs )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Absolute as Absolute ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import Algebra.Algebraic ( root )
+import qualified Algebra.RealField as RealField ( C )
+import qualified Algebra.ToInteger as ToInteger ( C )
+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 )
+
+
+-- | Instances of the 'Normed' class know how to compute their own
+--   p-norms for p=1,2,...,infinity.
+--
 class Normed a where
-  norm_p :: (Integral c, RealFrac b) => c -> a -> b
-  norm_infty :: RealFrac b => a -> b
+  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 2-norm.
+  norm :: (Algebraic.C b, Absolute.C b) => a -> b
+  norm = norm_p (2 :: Integer)
 
 -- Define instances for common numeric types.
 instance Normed Integer where
-  norm_p _ = fromInteger
-  norm_infty = fromInteger
+  norm_p _ = abs . fromInteger
+  norm_infty = abs . fromInteger
 
 instance Normed Rational where
-  norm_p _ = fromRational
-  norm_infty = fromRational
+  norm_p _ = abs . fromRational'
+  norm_infty = abs . fromRational'
 
 instance Epsilon e => Normed (BigFloat e) where
-  norm_p _ = fromRational . toRational
-  norm_infty = fromRational . toRational
+  norm_p _ = abs . fromRational' . toRational
+  norm_infty = abs . fromRational' . toRational
+
+instance Normed Float where
+  norm_p _ = abs . fromRational' . toRational
+  norm_infty = abs . fromRational' . toRational
 
 instance Normed Double where
-  norm_p _ = fromRational . toRational
-  norm_infty = fromRational . toRational
+  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