Split Roots into Roots.Simple and Roots.Fast.

[numerical-analysis.git] / src / Roots / Simple.hs

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
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+-- | The Roots.Simple module contains root-finding algorithms. That
+--   is, procedures to (numerically) find solutions to the equation,
+--
+--   > f(x) = 0
+--
+--   where f is assumed to be continuous on the interval of interest.
+--
+
+module Roots.Simple
+where
+
+import qualified Roots.Fast as F
+
+
+-- | 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
+--   and f(b) <= 0.
+--
+--   Examples:
+--
+--   >>> let f x = x**3
+--   >>> has_root f (-1) 1 Nothing
+--   True
+--
+--   This fails if we don't specify an @epsilon@, because cos(-2) ==
+--   cos(2) doesn't imply that there's a root on [-2,2].
+--
+--   >>> has_root cos (-2) 2 Nothing
+--   False
+--   >>> has_root cos (-2) 2 (Just 0.001)
+--   True
+--
+has_root :: (Fractional a, Ord a, Ord b, Num b)
+         => (a -> b) -- ^ The function @f@
+         -> a       -- ^ The \"left\" endpoint, @a@
+         -> a       -- ^ The \"right\" endpoint, @b@
+         -> Maybe a -- ^ The size of the smallest subinterval
+                   --   we'll examine, @epsilon@
+         -> Bool
+has_root f a b epsilon =
+  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.
+--
+--   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:
+--
+--   >>> bisect cos 1 2 0.001
+--   Just 1.5712890625
+--
+--   >>> bisect sin (-1) 1 0.001
+--   Just 0.0
+--
+bisect :: (Fractional a, Ord a, Num 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@
+       -> a       -- ^ The tolerance, @epsilon@
+       -> Maybe a
+bisect f a b epsilon =
+  F.bisect f a b epsilon Nothing Nothing