+
+
+-- | '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