Add a preliminary Piecewise module.

[numerical-analysis.git] / src / Piecewise.hs

diff --git a/src/Piecewise.hs b/src/Piecewise.hs
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+++ b/src/Piecewise.hs
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+{-# LANGUAGE ScopedTypeVariables #-}
+
+module Piecewise (
+  Piecewise(..),
+  evaluate,
+  evaluate',
+  from_intervals )
+where
+
+import qualified Algebra.Additive as Additive ( C )
+import Control.Arrow ( first )
+import NumericPrelude
+import qualified Prelude as P
+
+-- | A predicate is basically a function that returns True or
+--   False. In this case, the predicate is used to determine which
+--   piece of a 'Piecewise' function we want to use.
+type Predicate a = (a -> Bool)
+
+
+-- | One piece of a piecewise function. The predicate determines
+--   whether or not the point lies within this piece, and the second
+--   component (a function) is what we'll evaluate if it does.
+type Piece a = (Predicate a, (a -> a))
+
+-- | A representation of piecewise functions of one variable. A first
+--   approach might use a list of intervals paired with the function's
+--   value on those intervals. Rather than limit ourselves to a list
+--   of intervals, we allow the use of any predicate. The simple
+--   intervals case can be implemented with the predicate, \"is x
+--   between x1 and x2?\".
+data Piecewise a =
+  Piecewise { pieces :: [Piece a] }
+
+
+-- | Evaluate a piecewise function at a point. If the point is within
+--   the domain of the function, the function is evaluated at the
+--   point and the result is returned wrapped in a 'Just'. If the
+--   point is outside of the domain, 'Nothing' is returned.
+evaluate :: Piecewise a -> a -> Maybe a
+evaluate (Piecewise pieces) x =
+  foldl f Nothing pieces
+  where
+    f (Just y) _ = Just y
+    f Nothing (p,g) = if p x then Just (g x) else Nothing
+
+
+-- | Evaluate a piecewise function at a point. If the point is within
+--   the domain of the function, the function is evaluated at the
+--   point and the result is returned. If the point is outside of the
+--   domain, zero is returned.
+evaluate' :: (Additive.C a) => Piecewise a -> a -> a
+evaluate' pw x =
+  case evaluate pw x of
+    Nothing -> zero
+    Just result -> result
+
+
+-- | Construct a piecewise function from a list of intervals paired
+--   with functions. This is a convenience function that automatically
+--   converts intervals to \"contained in\" predicates.
+--
+--   Examples:
+--
+--   >>> let p1 = ((-1,0), \x -> -x) :: ((Double,Double), Double->Double)
+--   >>> let p2 = ((0,1), \x -> x) :: ((Double,Double), Double->Double)
+--   >>> let ivs = [p1, p2]
+--   >>> let pw = from_intervals ivs
+--   >>> evaluate' pw (-0.5)
+--   0.5
+--   >>> evaluate' pw 0.5
+--   0.5
+--
+from_intervals :: forall a. (Ord a) => [((a,a), (a -> a))] -> Piecewise a
+from_intervals =
+  Piecewise . map (first f)
+  where
+    f :: (a,a) -> Predicate a
+    f (x1,x2) x = x1 <= x && x <= x2