{-# LANGUAGE FlexibleInstances #-}
-
+{-# LANGUAGE RebindableSyntax #-}
-- | The 'Normed' class represents elements of a normed vector
-- space. We define instances for all common numeric types.
module Normed
where
-import Data.Number.BigFloat
+import BigFloat
+
+import NumericPrelude hiding (abs)
+import Algebra.Absolute
+import Algebra.Field
+import Algebra.Ring
+import Algebra.ToInteger
-- Since the norm is defined on a vector space, we should be able to
-- add and subtract anything on which a norm is defined. Of course
-- 'Num' is a bad choice here, but we really prefer to use the normal
-- addition and subtraction operators.
-class (Num a) => Normed a where
- norm_p :: (Integral c, RealFrac b) => c -> a -> b
- norm_infty :: RealFrac b => a -> b
+class (Algebra.Ring.C a, Algebra.Absolute.C a) => Normed a where
+ norm_p :: (Algebra.ToInteger.C c,
+ Algebra.Field.C b,
+ Algebra.Absolute.C b)
+ => c -> a -> b
+
+ norm_infty :: (Algebra.Field.C b,
+ Algebra.Absolute.C b)
+ => a -> b
-- | The "usual" norm. Defaults to the Euclidean norm.
- norm :: RealFrac b => a -> b
+ norm :: (Algebra.Field.C b, Algebra.Absolute.C b) => a -> b
norm = norm_p (2 :: Integer)
-- Define instances for common numeric types.
instance Normed Integer where
- norm_p _ = fromInteger
- norm_infty = fromInteger
+ norm_p _ = abs . fromInteger
+ norm_infty = abs . fromInteger
instance Normed Rational where
- norm_p _ = realToFrac
- norm_infty = realToFrac
+ norm_p _ = abs . fromRational'
+ norm_infty = abs . fromRational'
instance Epsilon e => Normed (BigFloat e) where
- norm_p _ = realToFrac
- norm_infty = realToFrac
+ norm_p _ = abs . fromRational' . toRational
+ norm_infty = abs . fromRational' . toRational
instance Normed Float where
- norm_p _ = realToFrac
- norm_infty = realToFrac
+ norm_p _ = abs . fromRational' . toRational
+ norm_infty = abs . fromRational' . toRational
instance Normed Double where
- norm_p _ = realToFrac
- norm_infty = realToFrac
+ norm_p _ = abs . fromRational' . toRational
+ norm_infty = abs . fromRational' . toRational
+{-# 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
import Normed
+import NumericPrelude hiding (abs)
+import Algebra.Absolute
+import Algebra.Field
+import Algebra.Ring
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+has_root :: (Algebra.Field.C a,
+ Ord a,
+ Algebra.Ring.C b,
+ Ord b,
+ Algebra.Absolute.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
c = (a + b)/2
-
-bisect :: (Fractional a, Ord a, Num b, Ord b)
+bisect :: (Algebra.Field.C a,
+ Ord a,
+ Algebra.Ring.C b,
+ Ord b,
+ Algebra.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@
+
-- | 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.
--
-- We also return the number of iterations required.
--
-fixed_point_with_iterations :: (Normed a, RealFrac b)
+fixed_point_with_iterations :: (Normed a,
+ Algebra.Field.C b,
+ Algebra.Absolute.C b,
+ Ord b)
=> (a -> a) -- ^ The function @f@ to iterate.
-> b -- ^ The tolerance, @epsilon@.
-> a -- ^ The initial value @x0@.
-- "safe" since the list is infinite. We'll succeed or loop
-- forever.
Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs
-
+{-# LANGUAGE RebindableSyntax #-}
+
-- | The Roots.Simple module contains root-finding algorithms. That
-- is, procedures to (numerically) find solutions to the equation,
--
import qualified Roots.Fast as F
+import NumericPrelude hiding (abs)
+import Algebra.Absolute
+import Algebra.Field
+import Algebra.Ring
+
-- | 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
-- [a,b] by the intermediate value theorem. Likewise when f(a) >= 0
-- >>> has_root cos (-2) 2 (Just 0.001)
-- True
--
-has_root :: (Fractional a, Ord a, Ord b, Num b)
+has_root :: (Algebra.Field.C a,
+ Ord a,
+ Algebra.Ring.C b,
+ Algebra.Absolute.C b,
+ Ord 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.
--
-- >>> bisect sin (-1) 1 0.001
-- Just 0.0
--
-bisect :: (Fractional a, Ord a, Num b, Ord b)
+bisect :: (Algebra.Field.C a,
+ Ord a,
+ Algebra.Ring.C b,
+ Algebra.Absolute.C b,
+ Ord 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
+-- | 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,
+ Algebra.Field.C b,
+ Algebra.Absolute.C b,
+ Ord 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,
+ Algebra.Field.C b,
+ Algebra.Absolute.C b,
+ Ord 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 :: (Normed a,
+ Algebra.Field.C b,
+ Algebra.Absolute.C b,
+ Ord 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
+
+
-- | The sequence x_{n} of values obtained by applying Newton's method
-- on the function @f@ and initial guess @x0@.
-- >>> tail $ take 4 $ newton_iterations f f' 2
-- [1.6806282722513088,1.4307389882390624,1.2549709561094362]
--
-newton_iterations :: (Fractional a, Ord a)
+newton_iterations :: (Algebra.Field.C a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
-> a -- ^ Initial guess, x-naught
xn - ( (f xn) / (f' xn) )
-
-- | Use Newton's method to find a root of @f@ near the initial guess
-- @x0@. If your guess is bad, this will recurse forever!
--
-- >>> abs (f root) < eps
-- True
--
-newtons_method :: (Fractional a, Ord a)
+newtons_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
-> a -- ^ The tolerance epsilon
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)...
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@.
--
-- >>> take 4 $ secant_iterations f 2 1
-- [2.0,1.0,1.0161290322580645,1.190577768676638]
--
-secant_iterations :: (Fractional a, Ord a)
+secant_iterations :: (Algebra.Field.C a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> a -- ^ Initial guess, x-naught
-> a -- ^ Second initial guess, x-one
-- >>> abs (f root) < (1/10^9)
-- True
--
-secant_method :: (Fractional a, Ord a)
+secant_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> a -- ^ The tolerance epsilon
-> a -- ^ Initial guess, x-naught
= find (\x -> abs (f x) < epsilon) x_n
where
x_n = secant_iterations f x0 x1
-
-
-
--- | 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, 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 =
- 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, 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
-
-
--- | 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 :: (Normed 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
projects / numerical-analysis.git / commitdiff