X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=79750e8d15b9aa2f0091e903b94a17accea9b7b8;hb=3a03a2bdc233f1504764b21149a13162486fc3bf;hp=0b0b93dfc05a070756a2838551eb0b3460f1f0e6;hpb=194f00d6448219fe1208ad59af136fbf67c863b5;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 0b0b93d..79750e8 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -114,6 +114,14 @@ 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@
@@ -202,3 +210,44 @@ 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, Ord a)
+            => (a -> a) -- ^ The function @f@ to iterate.
+            -> a       -- ^ 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 =
+      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
+
+    -- 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