X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=3237d60217aeb9e94bafe6983cf8f6a5dd353727;hb=c5da1efa77844ae6159dfc781ed886fdffbbf4d1;hp=7742355041e289f0a3f40ccdadbfa41dca5b6044;hpb=3bee9f6a68ae735d6c5084171e2fb521d1ce3f18;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 7742355..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
@@ -114,14 +116,22 @@ newton_iterations f f' x0 =
 --   >>> 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
 
@@ -205,41 +215,51 @@ secant_method f epsilon 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.
+--   at x0. We delegate to the version that returns the number of
+--   iterations and simply discard the number of iterations.
 --
-fixed_point :: (Num a, Ord a)
+fixed_point :: (Vector a, RealFrac b)
             => (a -> a) -- ^ The function @f@ to iterate.
-            -> a       -- ^ The tolerance, @epsilon@.
+            -> 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
+  snd $ F.fixed_point_with_iterations f epsilon x0
 
-    abs_diff v w =
-      abs (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
+-- | 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
 
-    -- 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 (\(x, diff) -> diff < epsilon) pairs
+-- | 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