{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE RebindableSyntax #-}
-- | Numerical solution of the initial value problem,
--
where
import Misc (partition)
-
+import NumericPrelude hiding (abs)
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.ToInteger as ToInteger
+import qualified Algebra.ToRational as ToRational
+import qualified Algebra.RealField as RealField
-- | A single iteration of Euler's method over the interval
-- [$x0$, $x0$+$h$].
-- >>> eulers_method1 x0 y0 f h
-- 2.0
--
-eulers_method1 :: (RealFrac a, RealFrac b)
+eulers_method1 :: (Field.C a, ToRational.C a, Field.C b)
=> a -- ^ x0, the initial point
-> b -- ^ y0, the initial value at x0
-> (a -> b -> b) -- ^ The function f(x,y)
eulers_method1 x0 y0 f h =
y0 + h'*y'
where
- h' = realToFrac h
+ h' = fromRational'$ toRational h
y' = (f x0 y0)
-- >>> head ys == y0
-- True
--
-eulers_method :: forall a b c. (RealFrac a, RealFrac b, Integral c)
+eulers_method :: forall a b c. (Field.C a,
+ ToRational.C a,
+ Field.C b,
+ ToInteger.C c,
+ Enum c)
=> a -- ^ x0, the initial point
-> b -- ^ y0, the initial value at x0
-> a -- ^ xN, the terminal point
-- >>> ys == ys'
-- True
--
-eulers_methodH :: (RealFrac a, RealFrac b)
+eulers_methodH :: (RealField.C a, ToRational.C a, Field.C b)
=> a -- ^ x0, the initial point
-> b -- ^ y0, the initial value at x0
-> a -- ^ xN, the terminal point
import Normed
import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
-
-has_root :: (Algebra.Field.C a,
- Ord a,
- Algebra.Ring.C b,
- Ord b,
- Algebra.Absolute.C b)
+import qualified Algebra.Absolute as Absolute
+import qualified Algebra.Field as Field
+import qualified Algebra.RealRing as RealRing
+import qualified Algebra.RealField as RealField
+
+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@
c = (a + b)/2
-bisect :: (Algebra.Field.C a,
- Ord a,
- Algebra.Ring.C b,
- Ord b,
- Algebra.Absolute.C 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@
-- We also return the number of iterations required.
--
fixed_point_with_iterations :: (Normed a,
- Algebra.Field.C b,
- Algebra.Absolute.C b,
+ Field.C b,
+ Absolute.C b,
Ord b)
=> (a -> a) -- ^ The function @f@ to iterate.
-> b -- ^ The tolerance, @epsilon@.
import qualified Roots.Fast as F
import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
+import qualified Algebra.Absolute as Absolute
+import Algebra.Absolute (abs)
+import qualified Algebra.Field as Field
+import qualified Algebra.RealField as RealField
+import qualified Algebra.RealRing as RealRing
-- | 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 :: (Algebra.Field.C a,
- Ord a,
- Algebra.Ring.C b,
- Algebra.Absolute.C b,
- Ord b)
+has_root :: (RealField.C a, RealRing.C b)
=> (a -> b) -- ^ The function @f@
-> a -- ^ The \"left\" endpoint, @a@
-> a -- ^ The \"right\" endpoint, @b@
-- >>> bisect sin (-1) 1 0.001
-- Just 0.0
--
-bisect :: (Algebra.Field.C a,
- Ord a,
- Algebra.Ring.C b,
- Algebra.Absolute.C 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@
-- 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)
+fixed_point :: (Normed a, RealField.C b)
=> (a -> a) -- ^ The function @f@ to iterate.
-> b -- ^ The tolerance, @epsilon@.
-> a -- ^ The initial value @x0@.
-- 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)
+fixed_point_iteration_count :: (Normed a, RealField.C b)
=> (a -> a) -- ^ The function @f@ to iterate.
-> b -- ^ The tolerance, @epsilon@.
-> a -- ^ The initial value @x0@.
--
-- 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)
+fixed_point_error_ratios :: (Normed a, RealField.C b)
=> (a -> a) -- ^ The function @f@ to iterate.
-> a -- ^ The initial value @x0@.
-> a -- ^ The true solution, @x_star@.
-- >>> tail $ take 4 $ newton_iterations f f' 2
-- [1.6806282722513088,1.4307389882390624,1.2549709561094362]
--
-newton_iterations :: (Algebra.Field.C a)
+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
-- >>> abs (f root) < eps
-- True
--
-newtons_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
+newtons_method :: (RealField.C a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
-> a -- ^ The tolerance epsilon
-- >>> take 4 $ secant_iterations f 2 1
-- [2.0,1.0,1.0161290322580645,1.190577768676638]
--
-secant_iterations :: (Algebra.Field.C a)
+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
-- >>> abs (f root) < (1/10^9)
-- True
--
-secant_method :: (Algebra.Field.C a, Algebra.Absolute.C a, Ord a)
+secant_method :: (RealField.C a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> a -- ^ The tolerance epsilon
-> a -- ^ Initial guess, x-naught
projects / numerical-analysis.git / commitdiff