X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=2906d95839c69daab27580863d8d61bb5f14ccc6;hb=5c0366134e8e1c12772cb685ac14b70d22d6ffed;hp=5aed7a1f4af210ea393033dd8bcf2f8b9c0f1923;hpb=28229a385d660108f9cc87e307512a07d6356b98;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 5aed7a1..2906d95 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -1,3 +1,6 @@
+{-# LANGUAGE RebindableSyntax #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+
 -- | The Roots.Simple module contains root-finding algorithms. That
 --   is, procedures to (numerically) find solutions to the equation,
 --
@@ -6,14 +9,34 @@
 --   where f is assumed to be continuous on the interval of interest.
 --
 
-module Roots.Simple
+module Roots.Simple (
+  bisect,
+  fixed_point,
+  fixed_point_error_ratios,
+  fixed_point_iteration_count,
+  has_root,
+  newtons_method,
+  secant_method,
+  trisect )
 where
 
 import Data.List (find)
+import NumericPrelude hiding ( abs )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import qualified Algebra.Field as Field ( C )
+import qualified Algebra.RealField as RealField ( C )
+import qualified Algebra.RealRing as RealRing ( C )
 
-import Vector
+import Normed ( Normed(..) )
+import qualified Roots.Fast as F (
+  bisect,
+  fixed_point_iterations,
+  fixed_point_with_iterations,
+  has_root,
+  trisect )
 
-import qualified Roots.Fast as F
 
 -- | 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
@@ -34,7 +57,7 @@ import qualified Roots.Fast as F
 --   >>> has_root cos (-2) 2 (Just 0.001)
 --   True
 --
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+has_root :: (RealField.C a, RealRing.C b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
          -> a       -- ^ The \"right\" endpoint, @b@
@@ -45,10 +68,8 @@ has_root f a b epsilon =
   F.has_root f a b epsilon Nothing Nothing
 
 
-
-
 -- | We are given a function @f@ and an interval [a,b]. The bisection
---   method checks finds a root by splitting [a,b] in half repeatedly.
+--   method finds a root by splitting [a,b] in half repeatedly.
 --
 --   If one is found within some prescribed tolerance @epsilon@, it is
 --   returned. Otherwise, the interval [a,b] is split into two
@@ -60,13 +81,17 @@ has_root f a b epsilon =
 --
 --   Examples:
 --
---   >>> bisect cos 1 2 0.001
---   Just 1.5712890625
+--   >>> let actual = 1.5707963267948966
+--   >>> let Just root = bisect cos 1 2 0.001
+--   >>> root
+--   1.5712890625
+--   >>> abs (root - actual) < 0.001
+--   True
 --
 --   >>> bisect sin (-1) 1 0.001
 --   Just 0.0
 --
-bisect :: (Fractional a, Ord a, Num 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@
@@ -76,6 +101,96 @@ bisect f a b epsilon =
   F.bisect f a b epsilon Nothing Nothing
 
 
+-- | We are given a function @f@ and an interval [a,b]. The trisection
+--   method finds a root by splitting [a,b] into three
+--   subintervals repeatedly.
+--
+--   If one is found within some prescribed tolerance @epsilon@, it is
+--   returned. Otherwise, the interval [a,b] is split into two
+--   subintervals [a,c] and [c,b] of equal length which are then both
+--   checked via the same process.
+--
+--   Returns 'Just' the value x for which f(x) == 0 if one is found,
+--   or Nothing if one of the preconditions is violated.
+--
+--   Examples:
+--
+--   >>> let actual = 1.5707963267948966
+--   >>> let Just root = trisect cos 1 2 0.001
+--   >>> root
+--   1.5713305898491083
+--   >>> abs (root - actual) < 0.001
+--   True
+--
+--   >>> trisect sin (-1) 1 0.001
+--   Just 0.0
+--
+trisect :: (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@
+       -> a       -- ^ The tolerance, @epsilon@
+       -> Maybe a
+trisect f a b epsilon =
+  F.trisect f a b epsilon Nothing Nothing
+
+
+-- | Find a fixed point of the function @f@ with the search starting
+--   at x0. We delegate to the version that returns the number of
+--   iterations and simply discard the number of iterations.
+--
+fixed_point :: (Normed a, Additive.C a, Algebraic.C b, RealField.C b)
+            => (a -> a) -- ^ The function @f@ to iterate.
+            -> b       -- ^ The tolerance, @epsilon@.
+            -> a       -- ^ The initial value @x0@.
+            -> a       -- ^ The fixed point.
+fixed_point f epsilon x0 =
+  snd $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | Return the number of iterations required to find a fixed point of
+--   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,
+                                Additive.C a,
+                                RealField.C b,
+                                Algebraic.C b)
+                            => (a -> a) -- ^ The function @f@ to iterate.
+                            -> b       -- ^ The tolerance, @epsilon@.
+                            -> a       -- ^ The initial value @x0@.
+                            -> Int       -- ^ The fixed point.
+fixed_point_iteration_count f epsilon x0 =
+  fst $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | Returns a list of ratios,
+--
+--     ||x^{*} - x_{n+1}|| / ||x^{*} - x_{n}||^{p}
+--
+--   of fixed point iterations for the function @f@ with initial guess
+--   @x0@ and @p@ some positive power.
+--
+--   This is used to determine the rate of convergence.
+--
+fixed_point_error_ratios :: forall a b. (Normed a,
+                             Additive.C a,
+                             RealField.C b,
+                             Algebraic.C b)
+                   => (a -> a) -- ^ The function @f@ to iterate.
+                   -> a       -- ^ The initial value @x0@.
+                   -> a       -- ^ The true solution, @x_star@.
+                   -> Integer -- ^ The power @p@.
+                   -> [b]     -- ^ The resulting sequence of x_{n}.
+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 (x_star - x)) xn :: [b]
+    en_plus_one = tail en
+    en_exp = map (^p) en
+
+
 
 -- | The sequence x_{n} of values obtained by applying Newton's method
 --   on the function @f@ and initial guess @x0@.
@@ -83,24 +198,24 @@ bisect f a b epsilon =
 --   Examples:
 --
 --   Atkinson, p. 60.
+--
 --   >>> let f x = x^6 - x - 1
 --   >>> let f' x = 6*x^5 - 1
 --   >>> tail $ take 4 $ newton_iterations f f' 2
 --   [1.6806282722513088,1.4307389882390624,1.2549709561094362]
 --
-newton_iterations :: (Fractional a, Ord 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
                     -> [a]
-newton_iterations f f' x0 =
-  iterate next x0
+newton_iterations f f' =
+  iterate next
   where
   next xn =
     xn - ( (f xn) / (f' xn) )
 
 
-
 -- | Use Newton's method to find a root of @f@ near the initial guess
 --   @x0@. If your guess is bad, this will recurse forever!
 --
@@ -124,7 +239,7 @@ newton_iterations f f' x0 =
 --   >>> abs (f root) < eps
 --   True
 --
-newtons_method :: (Fractional 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
@@ -136,7 +251,6 @@ newtons_method f f' epsilon x0 =
     x_n = newton_iterations f f' x0
 
 
-
 -- | Takes a function @f@ of two arguments and repeatedly applies @f@
 --   to the previous two values. Returns a list containing all
 --   generated values, f(x0, x1), f(x1, x2), f(x2, x3)...
@@ -158,6 +272,7 @@ iterate2 f x0 x1 =
       let next = f prev2 prev1 in
         next : go prev1 next
 
+
 -- | The sequence x_{n} of values obtained by applying the secant
 --   method on the function @f@ and initial guesses @x0@, @x1@.
 --
@@ -170,17 +285,18 @@ iterate2 f x0 x1 =
 --   Examples:
 --
 --   Atkinson, p. 67.
+--
 --   >>> let f x = x^6 - x - 1
 --   >>> take 4 $ secant_iterations f 2 1
 --   [2.0,1.0,1.0161290322580645,1.190577768676638]
 --
-secant_iterations :: (Fractional a, Ord 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
                     -> [a]
-secant_iterations f x0 x1 =
-  iterate2 g x0 x1
+secant_iterations f =
+  iterate2 g
   where
   g prev2 prev1 =
     let x_change = prev1 - prev2
@@ -195,6 +311,7 @@ secant_iterations f x0 x1 =
 --   Examples:
 --
 --   Atkinson, p. 67.
+--
 --   >>> let f x = x^6 - x - 1
 --   >>> let Just root = secant_method f (1/10^9) 2 1
 --   >>> root
@@ -202,7 +319,7 @@ secant_iterations f x0 x1 =
 --   >>> abs (f root) < (1/10^9)
 --   True
 --
-secant_method :: (Fractional 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
@@ -212,43 +329,3 @@ secant_method f epsilon x0 x1
   = find (\x -> abs (f x) < epsilon) x_n
   where
     x_n = secant_iterations f x0 x1
-
-
-
-fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
-                       -> a       -- ^ The initial value @x0@.
-                       -> [a]     -- ^ The resulting sequence of x_{n}.
-fixed_point_iterations f x0 =
-  iterate f x0
-
-
--- | Find a fixed point of the function @f@ with the search starting
---   at x0. This will find the first element in the chain f(x0),
---   f(f(x0)),... such that the magnitude of the difference between it
---   and the next element is less than epsilon.
---
-fixed_point :: (Num a, Vector a, RealFrac b)
-            => (a -> a) -- ^ The function @f@ to iterate.
-            -> b       -- ^ The tolerance, @epsilon@.
-            -> a       -- ^ The initial value @x0@.
-            -> a       -- ^ The fixed point.
-fixed_point f epsilon x0 =
-  fst winning_pair
-  where
-    xn = fixed_point_iterations f x0
-    xn_plus_one = tail $ fixed_point_iterations f x0
-
-    abs_diff v w = norm (v - w)
-
-    -- The nth entry in this list is the absolute value of x_{n} -
-    -- x_{n+1}.
-    differences = zipWith abs_diff xn xn_plus_one
-
-    -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
-    pairs = zip xn differences
-
-    -- The pair (xn, |x_{n} - x_{n+1}|) with
-    -- |x_{n} - x_{n+1}| < epsilon. The pattern match on 'Just' is
-    -- "safe" since the list is infinite. We'll succeed or loop
-    -- forever.
-    Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs