From: Michael Orlitzky Date: Sun, 3 Feb 2013 20:25:53 +0000 (-0500) Subject: Replace the custom 'Vector' typeclass with 'Normed' everywhere. X-Git-Url: http://gitweb.michael.orlitzky.com/?p=numerical-analysis.git;a=commitdiff_plain;h=e73e40c515938df4de629dbc88463c5d88bca7c8 Replace the custom 'Vector' typeclass with 'Normed' everywhere. --- diff --git a/src/FixedVector.hs b/src/FixedVector.hs index 6dabfb6..531c0fd 100644 --- a/src/FixedVector.hs +++ b/src/FixedVector.hs @@ -154,7 +154,7 @@ angle v1 v2 = acos theta where theta = (v1 `dot` v2) / norms - norms = (norm_p (2 :: Integer) v1) * (norm_p (2 :: Integer) v2) + norms = (norm v1) * (norm v2) -- | Convenience function for creating 2d vectors. diff --git a/src/Normed.hs b/src/Normed.hs index b60c2b1..7f4131a 100644 --- a/src/Normed.hs +++ b/src/Normed.hs @@ -7,10 +7,18 @@ where import Data.Number.BigFloat -class Normed a where +-- 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 + -- | The "usual" norm. Defaults to the Euclidean norm. + norm :: RealFrac b => a -> b + norm = norm_p (2 :: Integer) + -- Define instances for common numeric types. instance Normed Integer where norm_p _ = fromInteger diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs index cda999c..5efdf3b 100644 --- a/src/Roots/Fast.hs +++ b/src/Roots/Fast.hs @@ -8,7 +8,7 @@ where import Data.List (find) -import Vector +import Normed has_root :: (Fractional a, Ord a, Ord b, Num b) @@ -104,7 +104,7 @@ fixed_point_iterations f x0 = -- -- We also return the number of iterations required. -- -fixed_point_with_iterations :: (Vector a, RealFrac b) +fixed_point_with_iterations :: (Normed a, RealFrac b) => (a -> a) -- ^ The function @f@ to iterate. -> b -- ^ The tolerance, @epsilon@. -> a -- ^ The initial value @x0@. @@ -115,7 +115,7 @@ fixed_point_with_iterations f epsilon x0 = xn = fixed_point_iterations f x0 xn_plus_one = tail xn - abs_diff v w = norm_2 (v - w) + abs_diff v w = norm (v - w) -- The nth entry in this list is the absolute value of x_{n} - -- x_{n+1}. diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs index f9b36fa..6e3ff51 100644 --- a/src/Roots/Simple.hs +++ b/src/Roots/Simple.hs @@ -11,7 +11,7 @@ where import Data.List (find) -import Vector +import Normed import qualified Roots.Fast as F @@ -219,7 +219,7 @@ secant_method f epsilon x0 x1 -- at x0. We delegate to the version that returns the number of -- iterations and simply discard the number of iterations. -- -fixed_point :: (Vector a, RealFrac b) +fixed_point :: (Normed a, RealFrac b) => (a -> a) -- ^ The function @f@ to iterate. -> b -- ^ The tolerance, @epsilon@. -> a -- ^ The initial value @x0@. @@ -232,7 +232,7 @@ fixed_point f epsilon x0 = -- the function @f@ with the search starting at x0 and tolerance -- @epsilon@. We delegate to the version that returns the number of -- iterations and simply discard the fixed point. -fixed_point_iteration_count :: (Vector a, RealFrac b) +fixed_point_iteration_count :: (Normed a, RealFrac b) => (a -> a) -- ^ The function @f@ to iterate. -> b -- ^ The tolerance, @epsilon@. -> a -- ^ The initial value @x0@. @@ -250,7 +250,7 @@ fixed_point_iteration_count f epsilon x0 = -- -- This is used to determine the rate of convergence. -- -fixed_point_error_ratios :: (Vector a, RealFrac b) +fixed_point_error_ratios :: (Normed a, RealFrac b) => (a -> a) -- ^ The function @f@ to iterate. -> a -- ^ The initial value @x0@. -> a -- ^ The true solution, @x_star@. @@ -260,6 +260,6 @@ fixed_point_error_ratios f x0 x_star p = zipWith (/) en_plus_one en_exp where xn = F.fixed_point_iterations f x0 - en = map (\x -> norm_2 (x_star - x)) xn + en = map (\x -> norm (x_star - x)) xn en_plus_one = tail en en_exp = map (^p) en diff --git a/src/TwoTuple.hs b/src/TwoTuple.hs index 3950fee..becb11e 100644 --- a/src/TwoTuple.hs +++ b/src/TwoTuple.hs @@ -6,7 +6,7 @@ module TwoTuple where -import Vector +import Normed data TwoTuple a = TwoTuple a a @@ -18,10 +18,10 @@ instance (Show a) => Show (TwoTuple a) where instance Functor TwoTuple where f `fmap` (TwoTuple x1 y1) = TwoTuple (f x1) (f y1) -instance (RealFloat a) => Vector (TwoTuple a) where +instance (RealFloat a) => Normed (TwoTuple a) where -- The standard Euclidean 2-norm. We need RealFloat for the square -- root. - norm_2 (TwoTuple x y) = fromRational $ toRational (sqrt(x^2 + y^2)) + norm (TwoTuple x y) = fromRational $ toRational (sqrt(x^2 + y^2)) -- The infinity norm, i.e. the maximum entry. norm_infty (TwoTuple x y) = diff --git a/src/Vector.hs b/src/Vector.hs deleted file mode 100644 index 6953031..0000000 --- a/src/Vector.hs +++ /dev/null @@ -1,29 +0,0 @@ -{-# LANGUAGE FlexibleInstances #-} - --- | The 'Vector' class represents elements of a normed vector --- space. We define instances for all common numeric types. -module Vector -where - -import Data.Number.BigFloat - -class (Num a) => Vector a where - norm_2 :: RealFrac b => a -> b - norm_infty :: RealFrac b => a -> b - --- Define instances for common numeric types. -instance Vector Integer where - norm_2 = fromInteger - norm_infty = fromInteger - -instance Vector Rational where - norm_2 = fromRational - norm_infty = fromRational - -instance Epsilon e => Vector (BigFloat e) where - norm_2 = fromRational . toRational - norm_infty = fromRational . toRational - -instance Vector Double where - norm_2 = fromRational . toRational - norm_infty = fromRational . toRational