+{-# LANGUAGE RebindableSyntax #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+
-- | The Roots.Simple module contains root-finding algorithms. That
-- is, procedures to (numerically) find solutions to the equation,
--
-- where f is assumed to be continuous on the interval of interest.
--
-module Roots.Simple
+module Roots.Simple (
+ bisect,
+ fixed_point,
+ fixed_point_error_ratios,
+ fixed_point_iteration_count,
+ has_root,
+ newtons_method,
+ secant_method,
+ trisect )
where
import Data.List (find)
+import NumericPrelude hiding ( abs )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import qualified Algebra.Field as Field ( C )
+import qualified Algebra.RealField as RealField ( C )
+import qualified Algebra.RealRing as RealRing ( C )
+
+import Normed ( Normed(..) )
+import qualified Roots.Fast as F (
+ bisect,
+ fixed_point_iterations,
+ fixed_point_with_iterations,
+ has_root,
+ trisect )
-import qualified Roots.Fast as F
-- | Does the (continuous) function @f@ have a root on the interval
-- [a,b]? If f(a) <] 0 and f(b) ]> 0, we know that there's a root in
-- >>> has_root cos (-2) 2 (Just 0.001)
-- True
--
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+has_root :: (RealField.C a, RealRing.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
F.has_root f a b epsilon Nothing Nothing
-
-
-- | We are given a function @f@ and an interval [a,b]. The bisection
--- method checks finds a root by splitting [a,b] in half repeatedly.
+-- method finds a root by splitting [a,b] in half repeatedly.
--
-- If one is found within some prescribed tolerance @epsilon@, it is
-- returned. Otherwise, the interval [a,b] is split into two
--
-- Examples:
--
--- >>> bisect cos 1 2 0.001
--- Just 1.5712890625
+-- >>> let actual = 1.5707963267948966
+-- >>> let Just root = bisect cos 1 2 0.001
+-- >>> root
+-- 1.5712890625
+-- >>> abs (root - actual) < 0.001
+-- True
--
-- >>> bisect sin (-1) 1 0.001
-- Just 0.0
--
-bisect :: (Fractional a, Ord a, Num b, Ord b)
+bisect :: (RealField.C a, RealRing.C b)
=> (a -> b) -- ^ The function @f@ whose root we seek
-> a -- ^ The \"left\" endpoint of the interval, @a@
-> a -- ^ The \"right\" endpoint of the interval, @b@
F.bisect f a b epsilon Nothing Nothing
+-- | We are given a function @f@ and an interval [a,b]. The trisection
+-- method finds a root by splitting [a,b] into three
+-- subintervals repeatedly.
+--
+-- If one is found within some prescribed tolerance @epsilon@, it is
+-- returned. Otherwise, the interval [a,b] is split into two
+-- subintervals [a,c] and [c,b] of equal length which are then both
+-- checked via the same process.
+--
+-- Returns 'Just' the value x for which f(x) == 0 if one is found,
+-- or Nothing if one of the preconditions is violated.
+--
+-- Examples:
+--
+-- >>> let actual = 1.5707963267948966
+-- >>> let Just root = trisect cos 1 2 0.001
+-- >>> root
+-- 1.5713305898491083
+-- >>> abs (root - actual) < 0.001
+-- True
+--
+-- >>> trisect sin (-1) 1 0.001
+-- Just 0.0
+--
+trisect :: (RealField.C a, RealRing.C b)
+ => (a -> b) -- ^ The function @f@ whose root we seek
+ -> a -- ^ The \"left\" endpoint of the interval, @a@
+ -> a -- ^ The \"right\" endpoint of the interval, @b@
+ -> a -- ^ The tolerance, @epsilon@
+ -> Maybe a
+trisect f a b epsilon =
+ F.trisect f a b epsilon Nothing Nothing
+
+
+-- | Find a fixed point of the function @f@ with the search starting
+-- at x0. We delegate to the version that returns the number of
+-- iterations and simply discard the number of iterations.
+--
+fixed_point :: (Normed a, Additive.C a, Algebraic.C b, RealField.C 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 =
+ snd $ F.fixed_point_with_iterations f epsilon x0
+
+
+-- | 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 :: (Normed a,
+ Additive.C a,
+ RealField.C b,
+ Algebraic.C 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
+
+
+-- | 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 :: forall a b. (Normed a,
+ Additive.C a,
+ RealField.C b,
+ Algebraic.C 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 :: [b]
+ en_plus_one = tail en
+ en_exp = map (^p) en
+
+
-- | The sequence x_{n} of values obtained by applying Newton's method
-- on the function @f@ and initial guess @x0@.
-newton_iterations :: (Fractional a, Ord a)
+--
+-- 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 :: (Field.C a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
-> a -- ^ Initial guess, x-naught
-> [a]
-newton_iterations f f' x0 =
- iterate next x0
+newton_iterations f f' =
+ iterate next
where
next xn =
xn - ( (f xn) / (f' xn) )
-
-newtons_method :: (Fractional a, Ord a)
+-- | 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 :: (RealField.C 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
+
+
+-- | 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 :: (Field.C 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 =
+ iterate2 g
+ 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 :: (RealField.C 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
projects / numerical-analysis.git / blobdiff