import qualified Roots.Fast as F
import NumericPrelude hiding (abs)
-import Algebra.Absolute
-import Algebra.Field
-import Algebra.Ring
+import Algebra.Absolute (abs)
+import qualified Algebra.Additive as Additive
+import qualified Algebra.Algebraic as Algebraic
+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@
-- | 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 :: (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@
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,
- Algebra.Field.C b,
- Algebra.Absolute.C b,
- Ord b)
+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@.
-- @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)
+ 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@.
-- 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)
+ 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@.
-- >>> 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 / blobdiff