1 -- | The Roots.Simple module contains root-finding algorithms. That
2 -- is, procedures to (numerically) find solutions to the equation,
6 -- where f is assumed to be continuous on the interval of interest.
12 import Data.List (find)
14 import qualified Roots.Fast as F
16 -- | Does the (continuous) function @f@ have a root on the interval
17 -- [a,b]? If f(a) <] 0 and f(b) ]> 0, we know that there's a root in
18 -- [a,b] by the intermediate value theorem. Likewise when f(a) >= 0
24 -- >>> has_root f (-1) 1 Nothing
27 -- This fails if we don't specify an @epsilon@, because cos(-2) ==
28 -- cos(2) doesn't imply that there's a root on [-2,2].
30 -- >>> has_root cos (-2) 2 Nothing
32 -- >>> has_root cos (-2) 2 (Just 0.001)
35 has_root :: (Fractional a, Ord a, Ord b, Num b)
36 => (a -> b) -- ^ The function @f@
37 -> a -- ^ The \"left\" endpoint, @a@
38 -> a -- ^ The \"right\" endpoint, @b@
39 -> Maybe a -- ^ The size of the smallest subinterval
40 -- we'll examine, @epsilon@
42 has_root f a b epsilon =
43 F.has_root f a b epsilon Nothing Nothing
48 -- | We are given a function @f@ and an interval [a,b]. The bisection
49 -- method checks finds a root by splitting [a,b] in half repeatedly.
51 -- If one is found within some prescribed tolerance @epsilon@, it is
52 -- returned. Otherwise, the interval [a,b] is split into two
53 -- subintervals [a,c] and [c,b] of equal length which are then both
54 -- checked via the same process.
56 -- Returns 'Just' the value x for which f(x) == 0 if one is found,
57 -- or Nothing if one of the preconditions is violated.
61 -- >>> bisect cos 1 2 0.001
64 -- >>> bisect sin (-1) 1 0.001
67 bisect :: (Fractional a, Ord a, Num b, Ord b)
68 => (a -> b) -- ^ The function @f@ whose root we seek
69 -> a -- ^ The \"left\" endpoint of the interval, @a@
70 -> a -- ^ The \"right\" endpoint of the interval, @b@
71 -> a -- ^ The tolerance, @epsilon@
73 bisect f a b epsilon =
74 F.bisect f a b epsilon Nothing Nothing
78 -- | The sequence x_{n} of values obtained by applying Newton's method
79 -- on the function @f@ and initial guess @x0@.
80 newton_iterations :: (Fractional a, Ord a)
81 => (a -> a) -- ^ The function @f@ whose root we seek
82 -> (a -> a) -- ^ The derivative of @f@
83 -> a -- ^ Initial guess, x-naught
85 newton_iterations f f' x0 =
89 xn - ( (f xn) / (f' xn) )
93 newtons_method :: (Fractional a, Ord a)
94 => (a -> a) -- ^ The function @f@ whose root we seek
95 -> (a -> a) -- ^ The derivative of @f@
96 -> a -- ^ The tolerance epsilon
97 -> a -- ^ Initial guess, x-naught
99 newtons_method f f' epsilon x0
100 = find (\x -> abs (f x) < epsilon) x_n
102 x_n = newton_iterations f f' x0