{-# LANGUAGE RebindableSyntax #-}
+{-# LANGUAGE ScopedTypeVariables #-}
-- | 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
+import Normed ( Normed(..) )
+
+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 )
-import NumericPrelude hiding (abs)
-import qualified Algebra.Absolute as Absolute
-import qualified Algebra.Additive as Additive
-import qualified Algebra.Algebraic as Algebraic
-import qualified Algebra.RealRing as RealRing
-import qualified Algebra.RealField as RealField
has_root :: (RealField.C a,
RealRing.C 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
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
| f_of_b' == 0 = Just b
| otherwise =
-- Use a 'prime' just for consistency.
- let (a', b', fa', fb') =
- if (has_root f d b (Just epsilon) (Just f_of_d') (Just f_of_b'))
- then (d, b, f_of_d', f_of_b')
- else
- if (has_root f c d (Just epsilon) (Just f_of_c') (Just f_of_d'))
- then (c, d, f_of_c', f_of_d')
- else (a, c, f_of_a', f_of_c')
+ 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' = 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 = (2*a + b) / 3
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 :: (Normed a,
+fixed_point_with_iterations :: forall a b. (Normed a,
Additive.C a,
RealField.C b,
Algebraic.C b)
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
+ -- x_{n+1}. They're of type "b" because we're going to compare
+ -- them against epsilon.
+ differences = zipWith abs_diff xn xn_plus_one :: [b]
-- This produces the list [(n, xn)] so that we can determine
-- the number of iterations required.
- numbered_xn = zip [0..] xn
+ numbered_xn = zip [0..] xn :: [(Int,a)]
-- A list of pairs, (xn, |x_{n} - x_{n+1}|).
pairs = zip numbered_xn differences
projects / numerical-analysis.git / blobdiff