X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FRoots%2FSimple.hs;h=79750e8d15b9aa2f0091e903b94a17accea9b7b8;hb=3a03a2bdc233f1504764b21149a13162486fc3bf;hp=101b7e24c65610b381b9ff75d56137bc6f9b519f;hpb=587fda23c4c1e3f2f1a46063a9e6766d002ea356;p=numerical-analysis.git

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 101b7e2..79750e8 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -77,6 +77,15 @@ 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@
@@ -90,6 +99,29 @@ 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@
@@ -100,3 +132,122 @@ newtons_method f f' epsilon x0
   = find (\x -> abs (f x) < epsilon) x_n
   where
     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)...
+--
+--   Examples:
+--
+--   >>> let fibs = iterate2 (+) 0 1
+--   >>> take 15 fibs
+--   [0,1,1,2,3,5,8,13,21,34,55,89,144,233,377]
+--
+iterate2 :: (a -> a -> a) -- ^ The function @f@
+         -> a           -- ^ The initial value @x0@
+         -> a           -- ^ The second value, @x1@
+         -> [a]         -- ^ The result list, [x0, x1, ...]
+iterate2 f x0 x1 =
+  x0 : x1 : (go x0 x1)
+  where
+    go prev2 prev1 =
+      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@.
+--
+--   The recursion more or less implements a two-parameter 'iterate',
+--   although one list is passed to the next iteration (as opposed to
+--   one function argument, with iterate). At each step, we peel the
+--   first two elements off the list and then compute/append elements
+--   three, four... onto the end of the list.
+--
+--   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)
+                    => (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
+  where
+  g prev2 prev1 =
+    let x_change = prev1 - prev2
+        y_change = (f prev1) - (f prev2)
+    in
+      (prev1 - (f prev1 * (x_change / y_change)))
+
+
+-- | Use the secant method to find a root of @f@ near the initial guesses
+--   @x0@ and @x1@. If your guesses are bad, this will recurse forever!
+--
+--   Examples:
+--
+--   Atkinson, p. 67.
+--   >>> let f x = x^6 - x - 1
+--   >>> let Just root = secant_method f (1/10^9) 2 1
+--   >>> root
+--   1.1347241384015196
+--   >>> abs (f root) < (1/10^9)
+--   True
+--
+secant_method :: (Fractional a, Ord a)
+                 => (a -> a) -- ^ The function @f@ whose root we seek
+                 -> a       -- ^ The tolerance epsilon
+                 -> a       -- ^ Initial guess, x-naught
+                 -> a       -- ^ Second initial guess, x-one
+                 -> Maybe a
+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