From: Michael Orlitzky <michael@orlitzky.com>
Date: Mon, 24 Sep 2012 16:14:23 +0000 (-0400)
Subject: Add the secant method to Roots.Simple.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=b55c792bd9e2d439c5f1aebae160c92941a87e4e;p=numerical-analysis.git

Add the secant method to Roots.Simple.
---

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 101b7e2..2689163 100644
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -90,6 +90,8 @@ 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!
 newtons_method :: (Fractional a, Ord a)
                  => (a -> a) -- ^ The function @f@ whose root we seek
                  -> (a -> a) -- ^ The derivative of @f@
@@ -100,3 +102,81 @@ 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