X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FNormed.hs;h=28fbd2878a2d02641820081da67578ba1166bce3;hb=HEAD;hp=9bef7631221b9076fe57a261299c98147301233d;hpb=c3905924154d9a8d56bdc57e2f36fe48b8524eef;p=numerical-analysis.git

diff --git a/src/Normed.hs b/src/Normed.hs
index 9bef763..28fbd28 100644
--- a/src/Normed.hs
+++ b/src/Normed.hs
@@ -1,41 +1,91 @@
 {-# 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
 
--- 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 (Num a) => Normed a where
-  norm_p :: (Integral c, RealFrac b) => c -> a -> b
-  norm_infty :: RealFrac b => a -> b
+-- Ensure that we don't use the Lattice.C "max" function, that
+-- only works on Integer/Bool.
+import NumericPrelude hiding ( abs, max )
+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 qualified Data.Vector.Fixed as V (
+  Arity,
+  foldl,
+  map )
+import Data.Vector.Fixed.Boxed ( Vec )
+import qualified Prelude as P ( max )
+import Linear.Vector ( element_sum )
 
-  -- | The "usual" norm. Defaults to the Euclidean norm.
-  norm :: RealFrac b => a -> b
+
+-- | Instances of the 'Normed' class know how to compute their own
+--   p-norms for p=1,2,...,infinity.
+--
+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 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 _ = realToFrac
-  norm_infty = realToFrac
+  norm_p _ = abs . fromRational'
+  norm_infty = abs . fromRational'
 
 instance Epsilon e => Normed (BigFloat e) where
-  norm_p _ = realToFrac
-  norm_infty = realToFrac
+  norm_p _ = abs . fromRational' . toRational
+  norm_infty = abs . fromRational' . toRational
 
 instance Normed Float where
-  norm_p _ = realToFrac
-  norm_infty = realToFrac
+  norm_p _ = abs . fromRational' . toRational
+  norm_infty = abs . fromRational' . toRational
 
 instance Normed Double where
-  norm_p _ = realToFrac
-  norm_infty = realToFrac
+  norm_p _ = abs . fromRational' . toRational
+  norm_infty = abs . fromRational' . toRational
+
+
+-- | 'Normed' instance for vectors of any length. 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 (Vec0, 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
+--
+--   >>> let b = undefined :: Vec0 Int
+--   >>> norm b
+--   0.0
+--
+instance (V.Arity n, Absolute.C a, ToRational.C a, Ord a)
+         => Normed (Vec 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.foldl P.max 0) $ V.map abs x