X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=3237d60217aeb9e94bafe6983cf8f6a5dd353727;hb=c5da1efa77844ae6159dfc781ed886fdffbbf4d1;hp=2689163dbbe263ab0bb291958c652576a5797279;hpb=b55c792bd9e2d439c5f1aebae160c92941a87e4e;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 2689163..3237d60 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -11,6 +11,8 @@ where
 
 import Data.List (find)
 
+import Vector
+
 import qualified Roots.Fast as F
 
 -- | Does the (continuous) function @f@ have a root on the interval
@@ -77,6 +79,15 @@ bisect f a b epsilon =
 
 -- | The sequence x_{n} of values obtained by applying Newton's method
 --   on the function @f@ and initial guess @x0@.
+--
+--   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)
                     => (a -> a) -- ^ The function @f@ whose root we seek
                     -> (a -> a) -- ^ The derivative of @f@
@@ -92,14 +103,35 @@ newton_iterations f f' x0 =
 
 -- | Use Newton's method to find a root of @f@ near the initial guess
 --   @x0@. If your guess is bad, this will recurse forever!
+--
+--   Examples:
+--
+--   Atkinson, p. 60.
+--
+--   >>> let f x = x^6 - x - 1
+--   >>> let f' x = 6*x^5 - 1
+--   >>> let Just root = newtons_method f f' (1/1000000) 2
+--   >>> root
+--   1.1347241385002211
+--   >>> abs (f root) < 1/100000
+--   True
+--
+--   >>> import Data.Number.BigFloat
+--   >>> let eps = 1/(10^20) :: BigFloat Prec50
+--   >>> let Just root = newtons_method f f' eps 2
+--   >>> root
+--   1.13472413840151949260544605450647284028100785303643e0
+--   >>> abs (f root) < eps
+--   True
+--
 newtons_method :: (Fractional a, Ord a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> (a -> a) -- ^ The derivative of @f@
                  -> a       -- ^ The tolerance epsilon
                  -> a       -- ^ Initial guess, x-naught
                  -> Maybe a
-newtons_method f f' epsilon x0
-  = find (\x -> abs (f x) < epsilon) x_n
+newtons_method f f' epsilon x0 =
+  find (\x -> abs (f x) < epsilon) x_n
   where
     x_n = newton_iterations f f' x0
 
@@ -180,3 +212,54 @@ secant_method f epsilon x0 x1
   = find (\x -> abs (f x) < epsilon) x_n
   where
     x_n = secant_iterations f x0 x1
+
+
+
+-- | 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 :: (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 =
+  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 :: (Vector a, RealFrac 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 :: (Vector a, RealFrac 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
+    en_plus_one = tail en
+    en_exp = map (^p) en