Add an iteration count to the fixed_point function, rename it, and move it to the...

Implement fixed_point and fixed_point_iteration_count in Roots.Simple in terms of the general function.
Add a fixed_point_error_ratios function.

diff --git a/src/Roots/Fast.hs b/src/Roots/Fast.hs
index 8e49750e650a12f727d2c63282686bc0bfa7dfc1..d6e4d7c23f4091c7a573f01eb4d5c6572a1ca2c8 100644 (file)
--- a/src/Roots/Fast.hs
+++ b/src/Roots/Fast.hs
@@ -6,6 +6,11 @@
 module Roots.Fast
 where
 
+import Data.List (find)
+
+import Vector
+
+
 has_root :: (Fractional a, Ord a, Ord b, Num b)
          => (a -> b) -- ^ The function @f@
          -> a       -- ^ The \"left\" endpoint, @a@
@@ -80,3 +85,52 @@ bisect f a b epsilon f_of_a f_of_b
                  Just v -> v
 
     c = (a + b) / 2
+
+
+
+-- | Iterate the function @f@ with the initial guess @x0@ in hopes of
+--   finding a fixed point.
+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.
+--
+--   We also return the number of iterations required.
+--
+fixed_point_with_iterations :: (Vector a, RealFrac b)
+                            => (a -> a)  -- ^ The function @f@ to iterate.
+                            -> b        -- ^ The tolerance, @epsilon@.
+                            -> a        -- ^ The initial value @x0@.
+                            -> (Int, a) -- ^ The (iterations, fixed point) pair
+fixed_point_with_iterations f epsilon x0 =
+  (fst winning_pair)
+  where
+    xn = fixed_point_iterations f x0
+    xn_plus_one = tail xn
+
+    abs_diff v w = norm (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
+
+    -- This produces the list [(n, xn)] so that we can determine
+    -- the number of iterations required.
+    numbered_xn = zip [0..] xn
+
+    -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
+    pairs = zip numbered_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
+

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index 5aed7a1f4af210ea393033dd8bcf2f8b9c0f1923..3237d60217aeb9e94bafe6983cf8f6a5dd353727 100644 (file)
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -215,40 +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, Vector a, RealFrac b)
+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 =
-  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 = norm (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 (\(_, 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