X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=0a1debff0a0b1db653efabdf2be00ce9a4b91ed4;hb=59c49750fd2455574fe4e67ddd7e67a20321c8a8;hp=44d3d62112d2c4f339fa16b65c143e65ed5bb83a;hpb=fe73028041fe3becce6ce1ff268181d55d54a011;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 44d3d62..0a1debf 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -18,9 +18,11 @@ import Normed
 import qualified Roots.Fast as F
 
 import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
+import qualified Algebra.Absolute as Absolute
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.RealRing as RealRing
 
 -- | Does the (continuous) function @f@ have a root on the interval
 --   [a,b]? If f(a) <] 0 and f(b) ]> 0, we know that there's a root in
@@ -41,11 +43,7 @@ import Algebra.Ring
 --   >>> has_root cos (-2) 2 (Just 0.001)
 --   True
 --
-has_root :: (Algebra.Field.C a,
-             Ord a,
-             Algebra.Ring.C b,
-             Algebra.Absolute.C b,
-             Ord b)
+has_root :: (RealField.C a, RealRing.C b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
          -> a       -- ^ The \"right\" endpoint, @b@
@@ -75,11 +73,7 @@ has_root f a b epsilon =
 --   >>> bisect sin (-1) 1 0.001
 --   Just 0.0
 --
-bisect :: (Algebra.Field.C a,
-           Ord a,
-           Algebra.Ring.C b,
-           Algebra.Absolute.C b,
-           Ord b)
+bisect :: (RealField.C a, RealRing.C b)
        => (a -> b) -- ^ The function @f@ whose root we seek
        -> a       -- ^ The \"left\" endpoint of the interval, @a@
        -> a       -- ^ The \"right\" endpoint of the interval, @b@
@@ -93,10 +87,7 @@ bisect f a b epsilon =
 --   at x0. We delegate to the version that returns the number of
 --   iterations and simply discard the number of iterations.
 --
-fixed_point :: (Normed a,
-                Algebra.Field.C b,
-                Algebra.Absolute.C b,
-                Ord b)
+fixed_point :: (Normed a, RealField.C b)
             => (a -> a) -- ^ The function @f@ to iterate.
             -> b       -- ^ The tolerance, @epsilon@.
             -> a       -- ^ The initial value @x0@.
@@ -109,10 +100,7 @@ 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 :: (Normed a,
-                                Algebra.Field.C b,
-                                Algebra.Absolute.C b,
-                                Ord b)
+fixed_point_iteration_count :: (Normed a, RealField.C b)
                             => (a -> a) -- ^ The function @f@ to iterate.
                             -> b       -- ^ The tolerance, @epsilon@.
                             -> a       -- ^ The initial value @x0@.
@@ -130,10 +118,7 @@ fixed_point_iteration_count f epsilon x0 =
 --
 --   This is used to determine the rate of convergence.
 --
-fixed_point_error_ratios :: (Normed a,
-                             Algebra.Field.C b,
-                             Algebra.Absolute.C b,
-                             Ord b)
+fixed_point_error_ratios :: (Normed a, RealField.C b)
                    => (a -> a) -- ^ The function @f@ to iterate.
                    -> a       -- ^ The initial value @x0@.
                    -> a       -- ^ The true solution, @x_star@.
@@ -160,7 +145,7 @@ fixed_point_error_ratios f x0 x_star p =
 --   >>> tail $ take 4 $ newton_iterations f f' 2
 --   [1.6806282722513088,1.4307389882390624,1.2549709561094362]
 --
-newton_iterations :: (Algebra.Field.C a)
+newton_iterations :: (Field.C a)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> (a -> a) -- ^ The derivative of @f@
                     -> a       -- ^ Initial guess, x-naught
@@ -195,7 +180,7 @@ newton_iterations f f' x0 =
 --   >>> abs (f root) < eps
 --   True
 --
-newtons_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
+newtons_method :: (RealField.C a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> (a -> a) -- ^ The derivative of @f@
                  -> a       -- ^ The tolerance epsilon
@@ -245,7 +230,7 @@ iterate2 f x0 x1 =
 --   >>> take 4 $ secant_iterations f 2 1
 --   [2.0,1.0,1.0161290322580645,1.190577768676638]
 --
-secant_iterations :: (Algebra.Field.C a)
+secant_iterations :: (Field.C a)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> a       -- ^ Initial guess, x-naught
                     -> a       -- ^ Second initial guess, x-one
@@ -273,7 +258,7 @@ secant_iterations f x0 x1 =
 --   >>> abs (f root) < (1/10^9)
 --   True
 --
-secant_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
+secant_method :: (RealField.C a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> a       -- ^ The tolerance epsilon
                  -> a       -- ^ Initial guess, x-naught