+{-# LANGUAGE RebindableSyntax #-}
+
-- | The Roots.Fast module contains faster implementations of the
-- 'Roots.Simple' algorithms. Generally, we will pass precomputed
-- values to the next iteration of a function rather than passing
-- the function and the points at which to (re)evaluate it.
-module Roots.Fast
+module Roots.Fast (
+ bisect,
+ fixed_point_iterations,
+ fixed_point_with_iterations,
+ has_root,
+ trisect )
where
-import Data.List (find)
+import Data.List ( find )
+import Data.Maybe ( fromMaybe )
+
+import Normed ( Normed(..) )
-import Vector
+import NumericPrelude hiding ( abs )
+import qualified Algebra.Absolute as Absolute ( C )
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import qualified Algebra.RealRing as RealRing ( C )
+import qualified Algebra.RealField as RealField ( C )
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+has_root :: (RealField.C a,
+ RealRing.C b,
+ Absolute.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
-> Maybe b -- ^ Precoumpted f(a)
-> Maybe b -- ^ Precoumpted f(b)
-> Bool
-has_root f a b epsilon f_of_a f_of_b =
- if not ((signum (f_of_a')) * (signum (f_of_b')) == 1) then
- -- We don't care about epsilon here, there's definitely a root!
- True
- else
- if (b - a) <= epsilon' then
- -- Give up, return false.
- False
- else
- -- If either [a,c] or [c,b] have roots, we do too.
+has_root f a b epsilon f_of_a f_of_b
+ | (signum (f_of_a')) * (signum (f_of_b')) /= 1 = True
+ | (b - a) <= epsilon' = False
+ | otherwise =
(has_root f a c (Just epsilon') (Just f_of_a') Nothing) ||
(has_root f c b (Just epsilon') Nothing (Just f_of_b'))
where
-- If the size of the smallest subinterval is not specified,
-- assume we just want to check once on all of [a,b].
- epsilon' = case epsilon of
- Nothing -> (b-a)
- Just eps -> eps
+ epsilon' = fromMaybe (b-a) epsilon
-- Compute f(a) and f(b) only if needed.
- f_of_a' = case f_of_a of
- Nothing -> f a
- Just v -> v
-
- f_of_b' = case f_of_b of
- Nothing -> f b
- Just v -> v
+ f_of_a' = fromMaybe (f a) f_of_a
+ f_of_b' = fromMaybe (f b) f_of_b
c = (a + b)/2
-
-bisect :: (Fractional a, Ord a, Num b, Ord b)
+bisect :: (RealField.C a,
+ RealRing.C b,
+ Absolute.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@
else bisect f c b epsilon (Just f_of_c') (Just f_of_b')
where
-- Compute f(a) and f(b) only if needed.
- f_of_a' = case f_of_a of
- Nothing -> f a
- Just v -> v
-
- f_of_b' = case f_of_b of
- Nothing -> f b
- Just v -> v
+ f_of_a' = fromMaybe (f a) f_of_a
+ f_of_b' = fromMaybe (f b) f_of_b
c = (a + b) / 2
+trisect :: (RealField.C a,
+ RealRing.C b,
+ Absolute.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 b -- ^ Precomputed f(a)
+ -> Maybe b -- ^ Precomputed f(b)
+ -> Maybe a
+trisect f a b epsilon f_of_a f_of_b
+ -- We pass @epsilon@ to the 'has_root' function because if we want a
+ -- result within epsilon of the true root, we need to know that
+ -- there *is* a root within an interval of length epsilon.
+ | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
+ | f_of_a' == 0 = Just a
+ | f_of_b' == 0 = Just b
+ | otherwise =
+ -- Use a 'prime' just for consistency.
+ let (a', b', fa', fb')
+ | has_root f d b (Just epsilon) (Just f_of_d') (Just f_of_b') =
+ (d, b, f_of_d', f_of_b')
+ | has_root f c d (Just epsilon) (Just f_of_c') (Just f_of_d') =
+ (c, d, f_of_c', f_of_d')
+ | otherwise =
+ (a, c, f_of_a', f_of_c')
+ in
+ if (b-a) < 2*epsilon
+ then Just ((b+a)/2)
+ else trisect f a' b' epsilon (Just fa') (Just fb')
+ where
+ -- Compute f(a) and f(b) only if needed.
+ f_of_a' = fromMaybe (f a) f_of_a
+ f_of_b' = fromMaybe (f b) f_of_b
+
+ c = (2*a + b) / 3
+
+ d = (a + 2*b) / 3
+
+ f_of_c' = f c
+ f_of_d' = f d
+
+
+
-- | 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
+fixed_point_iterations =
+ iterate
-- | Find a fixed point of the function @f@ with the search starting
--
-- We also return the number of iterations required.
--
-fixed_point_with_iterations :: (Vector a, RealFrac b)
+fixed_point_with_iterations :: (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@.
xn = fixed_point_iterations f x0
xn_plus_one = tail xn
- abs_diff v w = norm_2 (v - w)
+ abs_diff v w = norm (v - w)
-- The nth entry in this list is the absolute value of x_{n} -
-- x_{n+1}.
-- "safe" since the list is infinite. We'll succeed or loop
-- forever.
Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs
-
projects / numerical-analysis.git / blobdiff